Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.48.0-wmf.1 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 0 34883 2844124 2839991 2026-11-03T07:14:37Z LokiZhan 3112588 Improved PCB layout explanations, corrected technical details, added illustrations, relevant Wikiversity links, and technical references. 2844124 wikitext text/x-wiki '''Printed Circuit Board Layout''' introduces the design of printed circuit boards (PCBs) for low-frequency [[Electronic Circuits|analog and digital circuits]]. It follows the process from an electronic schematic to a board ready for fabrication, including component placement, routing, layer selection, and manufacturing files. A schematic defines the circuit connections, but not trace dimensions, component spacing, return paths, or manufacturing clearances. These are determined during layout. == PCB Construction == A rigid PCB consists of an insulating laminate and one or more patterned copper layers. FR-4, a glass-reinforced epoxy laminate, is commonly used for general-purpose boards. [[File:Layers of Single-Layer PCB Simplified.svg|thumb|right|300px|Typical layers of a single-sided PCB, including the substrate, copper, solder mask, and silkscreen.]] Copper layers form the traces, pads, and other conductors. Boards with multiple copper layers can use plated holes to connect them. === Number of Layers === Single-sided boards contain one copper layer and are suitable for simple circuits with limited routing requirements. Double-sided boards use copper on both sides of the laminate. Plated-through holes and vias allow connections between the two layers. Multilayer boards contain additional copper layers separated by insulating materials. These layers may be assigned to signal routing, power distribution, or ground reference planes. Layer count depends on routing density, electrical requirements, mechanical constraints, and cost. Additional layers can provide room for routing and continuous reference planes, but increase fabrication complexity. === Copper Thickness === PCB copper is commonly specified by nominal weight in ounces per square foot (oz/ft²). One ounce per square foot is equivalent to approximately 35 µm (1.4 mils) of copper. On outer layers, plating may change the finished thickness from that of the starting foil.<ref name="pcbcool-copper">[https://pcbcool.com/technical-guides/high-current-pcb-design-guide/ High Current PCB Design Guide], PCBCool (manufacturer's technical guide). Discusses copper weight, trace dimensions, and fabrication limits.</ref> Trace width and copper thickness both affect conductor resistance, voltage drop, and temperature rise. The board's construction and cooling conditions also matter when sizing traces.<ref name="ipc2152">[https://www.ipc.org/TOC/IPC-2152.pdf IPC-2152: Standard for Determining Current Carrying Capacity in Printed Board Design], IPC, table of contents.</ref> === Holes and Vias === A plated-through hole has a conductive wall connecting copper layers and may also hold a component lead. A via connects copper features on different layers. Through vias extend through the board; blind and buried vias connect only selected layers. [[File:Via Types.svg|thumb|right|260px|Through, blind, and buried vias in a multilayer PCB.]] Non-plated holes are used for mounting or alignment where no electrical connection is needed. Hole size, finished diameter, annular ring dimensions, and spacing from nearby copper must be compatible with the selected fabrication process. === Solder Mask and Silkscreen === Solder mask covers selected copper surfaces and helps prevent unintended solder bridges during assembly. Openings in the solder mask expose pads and other features that require soldering or electrical contact. The silkscreen carries reference designators, polarity marks, and assembly information. Markings should be legible and kept clear of solderable pads. == PCB Design Workflow == Layout normally begins with a completed schematic. The following steps are a general sequence; changes may require returning to an earlier step. === Schematic Preparation === The schematic defines the electrical connections between components. It should include complete component references, pin assignments, power connections, and relevant component values. Before beginning layout, the schematic should be reviewed for missing connections, incorrect pin assignments, and other electrical errors. Electrical rule checking (ERC) can find unconnected pins and some conflicting pin types, but it cannot establish whether the circuit will work as intended. Component symbols must also be associated with the correct physical footprints before placement. === Footprint Selection === A footprint defines the pad and hole layout for a component. Choose footprints from the component manufacturer's package drawing and land-pattern recommendations. Electrically equivalent parts may have different packages or pin assignments. Important dimensions include pad spacing, lead pitch, hole diameter, package outline, and orientation. Particular care is required when creating custom footprints. Incorrect pad numbering or dimensions can make an otherwise correct circuit board unusable. 3D models may help with mechanical clearance checks, but dimensions should still be checked against the manufacturer's drawing. === Board Outline and Mechanical Constraints === The board outline defines the finished PCB's dimensions and shape. Mounting holes, connectors, switches, indicators, and other mechanically constrained features should be established before placing components that have no fixed location. Connector positions must be checked against mating cables, enclosures, and nearby components. Mounting holes require sufficient clearance for fasteners and assembly tools. The board edge should also have adequate clearance from copper features according to the fabrication requirements. === Component Placement === Component placement has a substantial influence on routing difficulty, electrical performance, and assembly accessibility. A common placement sequence is: # Position components with fixed mechanical locations, such as connectors, switches, and mounting holes. # Place major integrated circuits and other large components. # Position components that must remain electrically close to particular devices. # Arrange the remaining components to simplify routing and assembly. Place [[Electronics/Capacitors|decoupling capacitors]] close to the relevant power pins, keeping the supply and return paths short. This is especially important for circuits with fast current transients.<ref name="ti-layout">[https://www.ti.com/lit/an/slaa920/slaa920.pdf TAS2563 Schematic and Layout Guidelines], Texas Instruments, section 4, ''Decoupling Capacitors''.</ref> Oscillators and their associated components should follow the layout recommendations for the device concerned. Long traces and unnecessary routing changes around sensitive oscillator connections should be avoided. Allow room for soldering, inspection, and repairs. Consistent component orientation can help assembly, provided electrical and mechanical requirements are met. === Trace Routing === Routing connects the component pads with copper traces according to the schematic. In a single copper layer, unrelated traces cannot cross without creating a short circuit. On multilayer boards, traces may cross on different layers and connect between layers using vias. Routing should begin with connections that have significant electrical or physical constraints. These may include power conductors, sensitive analog signals, or connections that must remain short. General routing considerations include: * Use conductor widths appropriate for the expected current and allowable temperature rise. * Maintain sufficient clearance between unrelated copper features. * Avoid unnecessary vias and changes in routing direction. * Keep sensitive signal paths away from identified noise sources where practical. * Provide suitable return-current paths for signal and power connections. * Consider the effects of routing on component access and test-point placement. For power traces, check both voltage drop and temperature rise. Trace width alone does not determine current capacity; copper thickness and cooling conditions are also important.<ref name="ipc2152" /> High-speed and radio-frequency circuits may also require controlled impedance, which depends on trace geometry, dielectric materials, and nearby copper planes. === Power Distribution and Grounding === Power conductors should be sized according to their current requirements and the acceptable voltage drop. Ground connections carry return currents and affect noise and electromagnetic compatibility. A continuous ground plane can provide a low-impedance return path and reduce the need for individual ground traces. Splitting or interrupting a reference plane can force return currents to follow longer paths. Such interruptions should be considered when routing signals across different regions of a board. For mixed-signal circuits, plan placement and current-return paths before deciding whether separate ground regions are necessary. Copper pours may also be used to distribute current or provide additional conductive area. Their electrical connections and clearances should be checked after routing. == Design Rules == Design rules set the clearances and dimensions checked by PCB layout software. Typical rules include minimum trace width, conductor spacing, via dimensions, annular ring requirements, and copper-to-board-edge clearance. These values depend on the manufacturing capability selected for the design. A rule set suitable for one fabrication process may not be appropriate for another. Design rule checking (DRC) can identify clearance violations, unconnected items, and features that do not meet configured limits. It checks only the rules that have been set up, so a passing result does not guarantee a correct circuit or a manufacturable board.<ref name="kicad">[https://docs.kicad.org/master/en/pcbnew/pcbnew.html PCB Editor documentation], KiCad, sections on design rules checking and fabrication outputs.</ref> Engineers should review critical routing, component footprints, mechanical clearances, and manufacturing specifications in addition to automated checks. == Common Layout Problems == === Incorrect Footprints === An incorrect footprint may result in mismatched pad spacing, reversed pin assignments, or insufficient component clearance. The component datasheet and mechanical drawing should be checked before the layout is released. === Inadequate Trace Width === A narrow trace may cause excessive [[Ohm's law|voltage drop]] or temperature rise. Check the expected current and finished copper thickness, as well as the allowable temperature rise.<ref name="ipc2152" /> === Poor Decoupling Placement === A capacitor placed far from the IC adds impedance to the decoupling path. Keep the connection between the capacitor, supply pin, and return path short.<ref name="ti-layout" /> === Interrupted Return Paths === Signals crossing gaps in their reference planes may force return currents to take longer paths, increasing loop area and coupling. Check the return path when changing layers or routing near plane openings. === Insufficient Mechanical Clearance === Components positioned near mounting holes, connectors, or enclosure boundaries may interfere with assembly. Mechanical drawings and board models should be reviewed before production files are generated. == Preparing a Design for Manufacturing == After layout and DRC, generate the files needed for fabrication and, if applicable, assembly. Gerber files represent copper, solder mask, silkscreen, and other artwork layers. The fabrication data must also define the board outline and drilling information, commonly provided in separate drill files. Other exchange formats may be used when supported by the manufacturer.<ref name="ucamco">[https://www.ucamco.com/en/guest/downloads/gerber-format Gerber format specifications], Ucamco, the format's developer.</ref> [[File:Gerber-layers-example.png|thumb|right|220px|Example of separate Gerber layers for silkscreen, solder mask, and copper.]] A fabrication drawing or written specification may be needed for material, board thickness, copper weight, surface finish, and tolerances not fully described by the artwork files. For assembly, a bill of materials (BOM) and component placement data may also be required. Before sending the design to a fabricator, inspect the exported files in a Gerber viewer. Check the layer set, board outline, and alignment of copper features and holes against the original layout. This is a separate check from DRC.<ref name="kicad" /><ref name="ucamco" /> == Glossary == * '''Footprint:''' The physical pad, hole, and outline information associated with a component in PCB layout software. * '''Trace:''' A patterned copper conductor used to connect circuit features. * '''Via:''' A conductive interconnection between copper layers. * '''Copper plane:''' A relatively large continuous copper area used for electrical reference or current distribution. * '''Annular ring:''' The copper surrounding a drilled hole. * '''Solder mask:''' An insulating coating that leaves selected conductive features exposed. * '''Silkscreen:''' Printed identification and assembly markings on a PCB. * '''Stack-up:''' The arrangement and thicknesses of copper and insulating layers within a PCB. * '''DRC:''' Design rule checking, an automated process for identifying layout-rule violations. * '''Gerber:''' A widely used format for representing PCB manufacturing artwork. == References == <references /> 9io97i0xcp85rigkhaikdxnr6w9adld 0 60198 2843995 2843698 2026-11-02T17:09:49Z Weisongyang 3113400 Added Techniques, Ergonomics and Tips to learn typing in better way. 2843995 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} '''Typing''' is the process of entering or inputting text by pressing keys on a typewriter, computer keyboard, mobile phone, or calculator.<ref>[[Wikipedia: Typing]]</ref> Learners should practice typing for fifteen minutes each day until their typing speed is at least 30 words per minute. == Proper Technique == To type efficiently, learners should use the "touch typing" method. This involves resting your fingers on the "Home Row" (the keys A, S, D, F for the left hand and J, K, L, ; for the right hand). By keeping your fingers anchored here, you can learn to find all other keys without looking down at your hands. == Accuracy Before Speed == While reaching a speed of 30 words per minute (WPM) is a great beginner goal, accuracy is more important. It is better to type slowly and correctly than to type fast and make constant mistakes. Aim to maintain at least a 95% accuracy rate before trying to increase your speed. == Ergonomics == Good posture is essential for comfortable typing. Learners should sit up straight with their feet flat on the floor. Keep your wrists straight and hover them slightly above the keyboard rather than resting them on the desk while actively typing. == Practice Tips == * '''Don't look at your hands:''' Try to rely on muscle memory. If you make a mistake, feel for the correct key rather than looking down. * '''Use online tools:''' Free websites like TypingClub, Keybr, Monkeytype, Typingbattles offer interactive lessons to track your progress. * '''Take breaks:''' If your hands or wrists feel sore, take a short break and stretch your fingers. == Multimedia == * [https://www.typing.com/student/lessons Typing.com: Learn to Type] * [https://monkeytype.com/ Monkeytype] * [https://www.typingbattles.com/ TypingBattles] == Activities == * Complete the [https://www.typing.com/student/lessons Typing.com lessons.] * Use [https://monkeytype.com/ Monkeytype.com] , and practice typing daily until you can reach 30 words per minute consistently. == See Also == * [[Elementary Typing]] * [[Introduction_to_Computers/Input_Devices |Input Devices]] * [[Wikipedia: Typing]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category:Computer Skills]] [[Category:Typing]] [[Category:Completed resources]] fvaybj4666pdiugrnhi3ouvft1rkt8q 2844054 2843995 2026-11-02T23:00:45Z Atcovi 276019 Reverted edit by [[Special:Contributions/Weisongyang|Weisongyang]] ([[User_talk:Weisongyang|talk]]) to last version by [[User:Johannnes89|Johannnes89]] using [[Wikiversity:Rollback|rollback]] 2843698 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} '''Typing''' is the process of entering or inputting text by pressing keys on a typewriter, computer keyboard, mobile phone, or calculator.<ref>[[Wikipedia: Typing]]</ref> Learners should practice typing for fifteen minutes each day until their typing speed is at least 30 words per minute. == Multimedia == * [https://www.typing.com/student/lessons Typing.com: Learn to Type] * [https://monkeytype.com/ Monkeytype] == Activities == * Complete the [https://www.typing.com/student/lessons Typing.com lessons.] * Use [https://monkeytype.com/ Monkeytype.com] , and practice typing daily until you can reach 30 words per minute consistently. == See Also == * [[Elementary Typing]] * [[Introduction_to_Computers/Input_Devices |Input Devices]] * [[Wikipedia: Typing]] == References == {{reflist}} {{subpage navbar}} {{CourseCat}} [[Category:Computer Skills]] [[Category:Typing]] [[Category:Completed resources]] rmlo6eamspueufw9d1hcmsogmo5uq5p 0 98611 2844160 2815414 2026-11-03T10:18:22Z Jtneill 10242 /* Interventions */ Update experimental manipulations table to Reeve (2024) 2844160 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 12: Interventions and review|twelfth and final}} {{Motivation and emotion/Lectures/In development}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:Childcare Development Center-Crestwood High School cheerleaders 120815-F-PG936-400.jpg|right|250px|thumb|'''Figure 1'''. Interventions to change motivation and emotion are commonly used in educational settings.]] ==Overview== This lecture applies and reviews the [[motivation and emotion]] unit by discussing: * interventions using motivation and emotion principles * key points and wisdom gained Take-home messages: * Motivation guides engagement in behaviours that optimise well-being * Emotions provide feedback about progress towards goals * There is nothing so practical as a good theory ==Outline== * Interventions ** Interventions ** Practical problems ** Example interventions * Review ** Unit aims ** Nuggets of truth / Wisdom gained ** Review of key content ==Interventions== The ultimate test of a science is how well its principles can be applied to change desired outcomes. From a psychological point of view, motivation and emotion are dynamic and malleable and, therefore, can also be intentionally changed and strengthened. Well-designed interventions can develop motivational and emotional resources which help promote life outcomes that people care about (e.g., engagement, learning, performance, and well-being) (see Figure 1). The "trick" is not to try changing outcomes directly, but rather to ''alter the conditions'' so they are more conducive to the emergence of the desired outcomes. To systematically intervene, create and follow a plan: # '''Explain''': Diagnose–''Why'' is this happening? What is motivating this person? (Remember: ''All'' behaviour is motivated.) # '''Predict''': What will happen if nothing is done? What could happen if X changed? What if Y or Z changed? Consider options. # '''Intervene''': Apply motivation and emotion principles. Implement a step-by-step plan to alter: #* environment #* relationships #* motivation #* emotion # '''Observe/measure/adjust''': #* Measure key processes and outcomes (before, during, after) #* Track progress regularly #* Adjust strategies based on evidence of what works or not #* Document what was done, what changes, and why Change is difficult. Complex problems often need persistence, feedback, and refinement. If an intervention doesn't work, analyse why, and try a revised approach. <div align="center"> {| class="wikitable mw-collapsible" |+ Examples of brief experimental manipulations ! Ch ! Motivation/Emotion ! Experimental manipulation |- | 3 | Trust | Oxytocin hormone via nasal spray |- | 4 | Sexual attraction | Facial metrics (large eyes, small nose, small chin) |- | 5 | Extrinsic motivation | Offering a scholarship for high grades |- | 6 | Autonomy | Self-direction in pursuit of a personal goal |- | 7 | Discrepancy | A difficult, specific, self-concordant goal to strive for |- | 8 | Growth mindset | Story about how Einstein worked hard to become smart |- | 9 | Self-efficacy | A highly competent role model |- | 10 | Self-control depletion | Resisting an attractive temptation for 5 minutes |- | 11 | Positive affect | Receiving a small, unexpected gift |- | 12 | Sadness | Viewing a film about a son at his Dad’s funeral |- | 13 | Embarrassment | Committing a social blunder in front of an audience |- | 14 | Congruence | Unconditional positive regard |- | 15 | Obsession | Keep a secret about the person you are talking to |} </div> ''Note''. Based on Reeve (2024), Table 16.2, p. 350 Use theory to help explain the cause of what is happening. All theories are (potentially) "correct" — our job is to find the time and place when a theory can be usefully applied. <div align="center"> {| class="wikitable mw-collapsible" |+ Example mini-theories |- | Achievement goals || Arousal || Attribution |- | Broaden-and-build || Cognitive dissonance || Drive |- | Effectance motivation || Ego depletion || Ego development |- | Expectancy x Value || Facial feedback || Flow |- | Goal setting || Implicit motives || Learned helplessness |- | Mindsets || Positive affect || Psychodynamics |- | Reactance || Self-actualisation || Self-concordance |- | Self-determination || Self-efficacy || Stress and coping |} </div> ''Note''. Based on Reeve (2018, p. 424) What theories help predict motivational and/or emotional responses to the following conditions? * Deprivation from food and people * Seeing a threatening (angry) face * Smelling rotting food * Encountering an obstacle to one's goal * Expecting a tangible reward (e.g., money) * Uncontrollable environment * Failure in front of others * Observing a competent role model * Autonomy-supportive teacher There are two basic strategies for intervening: * Overcome: Fix what isn't working ** Repair weaknesses ** Overcome pathology * Promote ** Amplify strengths ** Improve functioning <div align="center"> {| class="wikitable mw-collapsible" |+ Basic intervention strategies ! Overcome !! Promote |- | Learned helplessness || Resilient self-efficacy |- | Hidden costs of reward || Autonomy need satisfaction |- | Fixed mindset || Difficult, specific, self-congruent goals |- | Depleted self-control || Mastery goals |- | Pessimistic explanatory style || Flow experience |- | Immature defense mechanisms || Organismic valuing process |- | Hubristic pride || Ego development |- | Malicious envy || Joy |- | || Gratitude |} </div> ''Note''. Based on Reeve (2018, pp. 425–426) ==Practical problems== Whilst these scenarios lack context, it should still be possible to generate some ideas about: * Why? (explain) * What will happen; but also, what could be? (predict) * Implementation (application) of strategies Explain → Predict → Intervene: * Child not wanting to brush teeth * Sales employee quota * Elite athlete performance * Weight loss * Study motivation These scenarios are explained in more detail below (based on Reeve, 2018, Box 17, p. 426) ;Child at home A 6-year girl resists brushing her teeth every night before bed. She finds the routine boring and doesn’t understand why it’s important. When asked to brush her teeth, she plays with the water and brushes for only a few seconds. Her parents, aware of the long-term benefits of good dental hygiene, find themselves in a constant battle to get her to complete the task properly. None of their strategies seem to work. This nightly conflict drains the family’s energy and patience. ; Employee at work A mid-level sales representative at a large company feels increasingly disillusioned with her role. Despite knowing she has the skills to sell, she faces rejection on nearly every call. Her company only measures her success through monthly quotas, providing no support for personal development or job satisfaction. She meets her quota some months, but the constant rejection and lack of recognition make her feel drained. Her confidence has eroded, and she is beginning to question whether this job aligns with her strengths and interests. She fantasises about quitting but worries about financial stability. ; Athlete or musician A promising 18-year-old athlete (or musician) is in love with her craft. She spends hours practicing, enjoys the discipline, and consistently performs well. Despite her dedication, her growth in skill level has stagnated over the past year. While her peers are progressing rapidly, she feels stuck. Coaches and teachers praise her current ability but offer no new insights on how to break through to the next level. Frustrated, she worries that no matter how hard she works, she won't reach her goal of becoming an elite performer. Her motivation remains high, but she fears plateauing permanently. ;Medical patient A middle-aged man has been advised by his physician to lose 15 kilograms to avoid serious health risks, including a possible heart attack. He acknowledges the logic of the advice but feels disconnected from the urgency of the situation. A long history of poor dietary habits and a sedentary lifestyle has left him with little confidence that he can make meaningful changes. Despite multiple attempts to exercise and eat healthier in the past, nothing has ever stuck. He feels overwhelmed by the prospect of overhauling his lifestyle and is pessimistic that the results would justify the effort. ;Suffering student A university student in his third year is taking an advanced course in a subject he thought he could handle. Despite attending lectures and completing the readings, he performed poorly on the mid-term exam, far below his usual standards. The grade was a shock, leaving him feeling embarrassed and ashamed. He begins to doubt his abilities and contemplates giving up entirely. Each new lecture feels more overwhelming, and his efforts in studying feel futile. As the second half of the semester looms, his motivation plummets, and he wishes he could just drop the course. ==Nuggets of truth== [[File:Pearl 1.svg|right|50px]] [[File:Treasurchest.svg|right|50px]] After a tsunami of theories and research, let's gather your pearls or wisdom or nuggets of truth about the psychology motivation and emotion: * What are the '''greatest insights''', '''significant learnings''', or '''best ideas''' you've acquired through this unit? * What was the '''take-home message'''(s) from the chapter you created? Contribute your insights to: * [[Motivation and emotion/Nuggets of truth|Nuggets of truth]] (Wikiversity) <!-- Example book chapter take-home messages: * [https://x.com/jtneill/status/1451068316131287041 X] (2021) * [[Motivation and emotion/Book/2011/Pearls_of_wisdom|Wikiversity]] (2011) --> ==Wisdom gained== {{Nutshell| Motivation and emotion fuel human behaviour, shaped by unconscious drives and conscious goals, with social relations playing a key role. While we may underestimate the power of biological urges and emotional dynamics, intrinsic motivation and clear plans unlock our best performance. Self-efficacy and self-control help balance short-term impulses with long-term aspirations. The development of emotional intelligence guides us toward growth and flourishing. Understanding the functional complexity of motivation and emotion can help us to thrive as fully functionally people.}} Wisdom gained from a scientific study of motivation and emotion includes that: # '''Motivation''' = behavioural energy, direction, and persistence # '''What we don't know''' about motivation and emotion exceeds what we do know # The '''brain''' is as much about motivation and emotion as it is about cognition and thinking # We underestimate how powerful a motivational force '''biological urges''' can be when not experiencing them # '''Quality''' of motivation (intrinsic vs. extrinsic) is important # To flourish, motivation needs supportive conditions, especially '''supportive relationships''' # '''Implicit''' (unconscious) '''motives''' predict behaviour better than explicit (conscious) motives # We do our best when we have a specific plan of action to pursue a '''difficult, specific, and self-congruent goal''' # People with '''different mindsets''' pursue goals in different ways # The core '''self-efficacy beliefs''' of "I can do it" and "it will work" underlie competent functioning # Exert '''self-control over short-term urges''' to effectively pursue long-term goals # All '''emotions are good''' because they serve a functional purpose # '''Other people''' are the source of most of our emotions # The more sophisticated our '''emotional repertoire''', the more likely we are to have the right emotions in every situation # '''Encouraging growth''' is more productive than trying to cure weakness # Motivation often arises from outside of '''conscious awareness''' # There is nothing so practical as a '''good theory''' ==Readings== * Chapter 17: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 16: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) ==Slides== * [https://docs.google.com/presentation/d/1rpaBEkSs5p0TFT3i5JT_o0vG3wZkTJtFAdMO74h8teQ/edit?usp=sharing Interventions] (Google Slides) * [https://docs.google.com/presentation/d/1kucigYPF-72cP0FAbzQZmnOUhNfilMSsbGOQnrnWRaM/edit?usp=sharing Review] (Google Slides) <!-- * [http://www.slideshare.net/jtneill/conclusion-and-review Lecture slides] (Slideshare, 2016) * Handouts **[[Media:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|100px]] * 2014 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] * 2011 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ==Recordings== ;13/11/2013 {{Motivation and emotion/Lectures/Recording3|2013|5b4fdfbe-0073-498f-ad85-53d9c6d06e89}} ;16/11/2011 {{Motivation and emotion/Lectures/Recording|2011|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.m4v|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.mp3}} <!-- * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.mp3 mp3] (audio) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.m4v m4v] (video) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872 echo360] (rich media) --> ==See also== * [[/Images/]] ; Lecture * [[{{#titleparts:{{PAGENAME}}|2}}/Growth psychology|Growth psychology]] (Previous lecture) ; Tutorial * [[Motivation and emotion/Tutorials/Review|Review]] (Tutorial) ==Recording== * [https://au-lti.bbcollab.com/recording/b13748d85c474a87a061ab86fdc19adc Lecture 12] (2025)<!-- * [https://au-lti.bbcollab.com/recording/daa1ba979ba045799efbe73f3226b00f Lecture 12] (2024) * [https://au-lti.bbcollab.com/recording/8ef2ff0521b1457197d7920a93cffc99 Lecture 12] (2023) * [https://au-lti.bbcollab.com/recording/53b16231d9904444bac31b14bc7784f6 Lecture 12 recording] (2022) * [https://au-lti.bbcollab.com/recording/0bd459db9ece46b388a4b794f5751cff Lecture 12 recording] (2021) * [https://au-lti.bbcollab.com/recording/e810c154cfbe41189f8fb4056c871206 Lecture 12 recording] (2020) * [https://echo360.org.au/media/ea2a5707-6be8-43fc-996c-e176ce54bc4f/public Lecture 12 recording] (2018) * [https://echo360.org.au/media/2417e774-f589-4635-9955-9aaf73bcd3b8/public Lecture 12 recording] (2017)--> ==References== {{Hanging indent|1= Cheon, S. H., Reeve, J., & Moon, I. S. (2012). Experimentally based, longitudinally designed, teacher-focused intervention to help physical education teachers be more autonomy supportive toward their students. ''Journal of Sport and Exercise Psychology'', ''34''(3), 365–396. https://doi.org/10.1123/jsep.34.3.365 Izard, C. E., King, K. A., Trentacosta, C. J., Morgan, J. K., Laurenceau, J.-P., Krauthamer-Ewing, S. E., & Finlon, K. J. (2008). Accelerating the development of emotion competence in Head Start children: Effects on adaptive and maladaptive behavior. ''Development and Psychopathology'', ''20'', 369–397. https://doi.org/10.1017/S0954579408000175[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/82060B84CF144DBC4E1AC53CEC26A494/S0954579408000175a.pdf/accelerating_the_development_of_emotion_competence_in_head_start_children_effects_on_adaptive_and_maladaptive_behavior.pdf] Yeager, D. S., Trzesniewski, K. H., & Dweck, C. S. (2013). An implicit theory of personality intervention reduces adolescent aggression in response to victimization and exclusion. ''Child Development'', ''84'', 970–988. https://doi.org/10.1111/cdev.12003 }} <!-- ==External links== --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Interventions and review]] 1iqnz059tp9ra4q53tpn1a76ow1dap7 2844161 2844160 2026-11-03T10:34:37Z Jtneill 10242 /* Interventions */ + sub-headings 2844161 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 12: Interventions and review|twelfth and final}} {{Motivation and emotion/Lectures/In development}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:Childcare Development Center-Crestwood High School cheerleaders 120815-F-PG936-400.jpg|right|250px|thumb|'''Figure 1'''. Interventions to change motivation and emotion are commonly used in educational settings.]] ==Overview== This lecture applies and reviews the [[motivation and emotion]] unit by discussing: * interventions using motivation and emotion principles * key points and wisdom gained Take-home messages: * Motivation guides engagement in behaviours that optimise well-being * Emotions provide feedback about progress towards goals * There is nothing so practical as a good theory ==Outline== * Interventions ** Interventions ** Practical problems ** Example interventions * Review ** Unit aims ** Nuggets of truth / Wisdom gained ** Review of key content ==Interventions== The ultimate test of a science is how well its principles can be applied to change desired outcomes. From a psychological point of view, motivation and emotion are dynamic and malleable and, therefore, can also be intentionally changed and strengthened. Well-designed interventions can develop motivational and emotional resources which help promote life outcomes that people care about (e.g., engagement, learning, performance, and well-being) (see Figure 1). The "trick" is not to try changing outcomes directly, but rather to ''alter the conditions'' so they are more conducive to the emergence of the desired outcomes. ==Plan== To systematically intervene, create and follow a plan: # '''Explain''': Diagnose–''Why'' is this happening? What is motivating this person? (Remember: ''All'' behaviour is motivated.) # '''Predict''': What will happen if nothing is done? What could happen if X changed? What if Y or Z changed? Consider options. # '''Intervene''': Apply motivation and emotion principles. Implement a step-by-step plan to alter: #* environment #* relationships #* motivation #* emotion # '''Observe/measure/adjust''': #* Measure key processes and outcomes (before, during, after) #* Track progress regularly #* Adjust strategies based on evidence of what works or not #* Document what was done, what changes, and why Change is difficult. Complex problems often need persistence, feedback, and refinement. If an intervention doesn't work, analyse why, and try a revised approach. ==Experimental manipulations== <div align="center"> {| class="wikitable mw-collapsible" |+ Examples of brief experimental manipulations ! Ch ! Motivation/Emotion ! Experimental manipulation |- | 3 | Trust | Oxytocin hormone via nasal spray |- | 4 | Sexual attraction | Facial metrics (large eyes, small nose, small chin) |- | 5 | Extrinsic motivation | Offering a scholarship for high grades |- | 6 | Autonomy | Self-direction in pursuit of a personal goal |- | 7 | Discrepancy | A difficult, specific, self-concordant goal to strive for |- | 8 | Growth mindset | Story about how Einstein worked hard to become smart |- | 9 | Self-efficacy | A highly competent role model |- | 10 | Self-control depletion | Resisting an attractive temptation for 5 minutes |- | 11 | Positive affect | Receiving a small, unexpected gift |- | 12 | Sadness | Viewing a film about a son at his Dad’s funeral |- | 13 | Embarrassment | Committing a social blunder in front of an audience |- | 14 | Congruence | Unconditional positive regard |- | 15 | Obsession | Keep a secret about the person you are talking to |} </div> ''Note''. Based on Reeve (2024), Table 16.2, p. 350 ==Mini-theories== Use theory to help explain the cause of what is happening and possible forms of intervention. All theories are (potentially) "correct"—the practitioner's job is to match the moment (time, place, context) with a potentially useful mini-theory. Here's a list of some key mini-theories discussed in this unit: * Achievement motivation * Arousal * Attribution * Basic psychological needs * Broaden-and-build * Cognitive dissonance * Cognitive evaluation theory * Drive * Ego depletion * Ego development * Facial feedback * Flow * Fully functioning person * Goal setting * Intrinsic-extrinsic motivation * Learned helplessness * Mindsets * Positive affect * Psychodynamics * Reactance * Self-actualisation * Self-concordance * Self-determination * Self-efficacy * Stress and coping What theories could help predict motivational and/or emotional responses to the following conditions? * Deprivation from food and people * Seeing a threatening (angry) face * Smelling rotting food * Encountering an obstacle to one's goal * Expecting a tangible reward (e.g., money) * Uncontrollable environment * Failure in front of others * Observing a competent role model * Autonomy-supportive teacher ==Intervention strategies== There are two basic strategies for intervening: * Overcome: Fix what isn't working ** Repair weaknesses ** Overcome pathology * Promote and further develop what is working ** Amplify strengths ** Improve functioning <div align="center"> {| class="wikitable mw-collapsible" |+ Basic intervention strategies ! Overcome !! Promote |- | Learned helplessness || Resilient self-efficacy |- | Hidden costs of reward || Autonomy need satisfaction |- | Fixed mindset || Difficult, specific, self-congruent goals |- | Depleted self-control || Mastery goals |- | Pessimistic explanatory style || Flow experience |- | Immature defense mechanisms || Organismic valuing process |- | Hubristic pride || Ego development |- | Malicious envy || Joy |- | || Gratitude |} </div> ''Note''. Based on Reeve (2018, pp. 425–426) ==Practical problems== Whilst these scenarios lack context, it should still be possible to generate some ideas about: * Why? (explain) * What will happen; but also, what could be? (predict) * Implementation (application) of strategies Explain → Predict → Intervene: * Child not wanting to brush teeth * Sales employee quota * Elite athlete performance * Weight loss * Study motivation These scenarios are explained in more detail below (based on Reeve, 2018, Box 17, p. 426) ;Child at home A 6-year girl resists brushing her teeth every night before bed. She finds the routine boring and doesn’t understand why it’s important. When asked to brush her teeth, she plays with the water and brushes for only a few seconds. Her parents, aware of the long-term benefits of good dental hygiene, find themselves in a constant battle to get her to complete the task properly. None of their strategies seem to work. This nightly conflict drains the family’s energy and patience. ; Employee at work A mid-level sales representative at a large company feels increasingly disillusioned with her role. Despite knowing she has the skills to sell, she faces rejection on nearly every call. Her company only measures her success through monthly quotas, providing no support for personal development or job satisfaction. She meets her quota some months, but the constant rejection and lack of recognition make her feel drained. Her confidence has eroded, and she is beginning to question whether this job aligns with her strengths and interests. She fantasises about quitting but worries about financial stability. ; Athlete or musician A promising 18-year-old athlete (or musician) is in love with her craft. She spends hours practicing, enjoys the discipline, and consistently performs well. Despite her dedication, her growth in skill level has stagnated over the past year. While her peers are progressing rapidly, she feels stuck. Coaches and teachers praise her current ability but offer no new insights on how to break through to the next level. Frustrated, she worries that no matter how hard she works, she won't reach her goal of becoming an elite performer. Her motivation remains high, but she fears plateauing permanently. ;Medical patient A middle-aged man has been advised by his physician to lose 15 kilograms to avoid serious health risks, including a possible heart attack. He acknowledges the logic of the advice but feels disconnected from the urgency of the situation. A long history of poor dietary habits and a sedentary lifestyle has left him with little confidence that he can make meaningful changes. Despite multiple attempts to exercise and eat healthier in the past, nothing has ever stuck. He feels overwhelmed by the prospect of overhauling his lifestyle and is pessimistic that the results would justify the effort. ;Suffering student A university student in his third year is taking an advanced course in a subject he thought he could handle. Despite attending lectures and completing the readings, he performed poorly on the mid-term exam, far below his usual standards. The grade was a shock, leaving him feeling embarrassed and ashamed. He begins to doubt his abilities and contemplates giving up entirely. Each new lecture feels more overwhelming, and his efforts in studying feel futile. As the second half of the semester looms, his motivation plummets, and he wishes he could just drop the course. ==Nuggets of truth== [[File:Pearl 1.svg|right|50px]] [[File:Treasurchest.svg|right|50px]] After a tsunami of theories and research, let's gather your pearls or wisdom or nuggets of truth about the psychology motivation and emotion: * What are the '''greatest insights''', '''significant learnings''', or '''best ideas''' you've acquired through this unit? * What was the '''take-home message'''(s) from the chapter you created? Contribute your insights to: * [[Motivation and emotion/Nuggets of truth|Nuggets of truth]] (Wikiversity) <!-- Example book chapter take-home messages: * [https://x.com/jtneill/status/1451068316131287041 X] (2021) * [[Motivation and emotion/Book/2011/Pearls_of_wisdom|Wikiversity]] (2011) --> ==Wisdom gained== {{Nutshell| Motivation and emotion fuel human behaviour, shaped by unconscious drives and conscious goals, with social relations playing a key role. While we may underestimate the power of biological urges and emotional dynamics, intrinsic motivation and clear plans unlock our best performance. Self-efficacy and self-control help balance short-term impulses with long-term aspirations. The development of emotional intelligence guides us toward growth and flourishing. Understanding the functional complexity of motivation and emotion can help us to thrive as fully functionally people.}} Wisdom gained from a scientific study of motivation and emotion includes that: # '''Motivation''' = behavioural energy, direction, and persistence # '''What we don't know''' about motivation and emotion exceeds what we do know # The '''brain''' is as much about motivation and emotion as it is about cognition and thinking # We underestimate how powerful a motivational force '''biological urges''' can be when not experiencing them # '''Quality''' of motivation (intrinsic vs. extrinsic) is important # To flourish, motivation needs supportive conditions, especially '''supportive relationships''' # '''Implicit''' (unconscious) '''motives''' predict behaviour better than explicit (conscious) motives # We do our best when we have a specific plan of action to pursue a '''difficult, specific, and self-congruent goal''' # People with '''different mindsets''' pursue goals in different ways # The core '''self-efficacy beliefs''' of "I can do it" and "it will work" underlie competent functioning # Exert '''self-control over short-term urges''' to effectively pursue long-term goals # All '''emotions are good''' because they serve a functional purpose # '''Other people''' are the source of most of our emotions # The more sophisticated our '''emotional repertoire''', the more likely we are to have the right emotions in every situation # '''Encouraging growth''' is more productive than trying to cure weakness # Motivation often arises from outside of '''conscious awareness''' # There is nothing so practical as a '''good theory''' ==Readings== * Chapter 17: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 16: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) ==Slides== * [https://docs.google.com/presentation/d/1rpaBEkSs5p0TFT3i5JT_o0vG3wZkTJtFAdMO74h8teQ/edit?usp=sharing Interventions] (Google Slides) * [https://docs.google.com/presentation/d/1kucigYPF-72cP0FAbzQZmnOUhNfilMSsbGOQnrnWRaM/edit?usp=sharing Review] (Google Slides) <!-- * [http://www.slideshare.net/jtneill/conclusion-and-review Lecture slides] (Slideshare, 2016) * Handouts **[[Media:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|100px]] * 2014 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] * 2011 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ==Recordings== ;13/11/2013 {{Motivation and emotion/Lectures/Recording3|2013|5b4fdfbe-0073-498f-ad85-53d9c6d06e89}} ;16/11/2011 {{Motivation and emotion/Lectures/Recording|2011|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.m4v|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.mp3}} <!-- * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.mp3 mp3] (audio) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.m4v m4v] (video) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872 echo360] (rich media) --> ==See also== * [[/Images/]] ; Lecture * [[{{#titleparts:{{PAGENAME}}|2}}/Growth psychology|Growth psychology]] (Previous lecture) ; Tutorial * [[Motivation and emotion/Tutorials/Review|Review]] (Tutorial) ==Recording== * [https://au-lti.bbcollab.com/recording/b13748d85c474a87a061ab86fdc19adc Lecture 12] (2025)<!-- * [https://au-lti.bbcollab.com/recording/daa1ba979ba045799efbe73f3226b00f Lecture 12] (2024) * [https://au-lti.bbcollab.com/recording/8ef2ff0521b1457197d7920a93cffc99 Lecture 12] (2023) * [https://au-lti.bbcollab.com/recording/53b16231d9904444bac31b14bc7784f6 Lecture 12 recording] (2022) * [https://au-lti.bbcollab.com/recording/0bd459db9ece46b388a4b794f5751cff Lecture 12 recording] (2021) * [https://au-lti.bbcollab.com/recording/e810c154cfbe41189f8fb4056c871206 Lecture 12 recording] (2020) * [https://echo360.org.au/media/ea2a5707-6be8-43fc-996c-e176ce54bc4f/public Lecture 12 recording] (2018) * [https://echo360.org.au/media/2417e774-f589-4635-9955-9aaf73bcd3b8/public Lecture 12 recording] (2017)--> ==References== {{Hanging indent|1= Cheon, S. H., Reeve, J., & Moon, I. S. (2012). Experimentally based, longitudinally designed, teacher-focused intervention to help physical education teachers be more autonomy supportive toward their students. ''Journal of Sport and Exercise Psychology'', ''34''(3), 365–396. https://doi.org/10.1123/jsep.34.3.365 Izard, C. E., King, K. A., Trentacosta, C. J., Morgan, J. K., Laurenceau, J.-P., Krauthamer-Ewing, S. E., & Finlon, K. J. (2008). Accelerating the development of emotion competence in Head Start children: Effects on adaptive and maladaptive behavior. ''Development and Psychopathology'', ''20'', 369–397. https://doi.org/10.1017/S0954579408000175[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/82060B84CF144DBC4E1AC53CEC26A494/S0954579408000175a.pdf/accelerating_the_development_of_emotion_competence_in_head_start_children_effects_on_adaptive_and_maladaptive_behavior.pdf] Yeager, D. S., Trzesniewski, K. H., & Dweck, C. S. (2013). An implicit theory of personality intervention reduces adolescent aggression in response to victimization and exclusion. ''Child Development'', ''84'', 970–988. https://doi.org/10.1111/cdev.12003 }} <!-- ==External links== --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Interventions and review]] fs09sabe6dyt6c4htjuky1u01hw0d3w 2844164 2844161 2026-11-03T10:43:35Z Jtneill 10242 /* Intervention strategies */ Remove confusing overcome/promote table 2844164 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 12: Interventions and review|twelfth and final}} {{Motivation and emotion/Lectures/In development}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:Childcare Development Center-Crestwood High School cheerleaders 120815-F-PG936-400.jpg|right|250px|thumb|'''Figure 1'''. Interventions to change motivation and emotion are commonly used in educational settings.]] ==Overview== This lecture applies and reviews the [[motivation and emotion]] unit by discussing: * interventions using motivation and emotion principles * key points and wisdom gained Take-home messages: * Motivation guides engagement in behaviours that optimise well-being * Emotions provide feedback about progress towards goals * There is nothing so practical as a good theory ==Outline== * Interventions ** Interventions ** Practical problems ** Example interventions * Review ** Unit aims ** Nuggets of truth / Wisdom gained ** Review of key content ==Interventions== The ultimate test of a science is how well its principles can be applied to change desired outcomes. From a psychological point of view, motivation and emotion are dynamic and malleable and, therefore, can also be intentionally changed and strengthened. Well-designed interventions can develop motivational and emotional resources which help promote life outcomes that people care about (e.g., engagement, learning, performance, and well-being) (see Figure 1). The "trick" is not to try changing outcomes directly, but rather to ''alter the conditions'' so they are more conducive to the emergence of the desired outcomes. ==Plan== To systematically intervene, create and follow a plan: # '''Explain''': Diagnose–''Why'' is this happening? What is motivating this person? (Remember: ''All'' behaviour is motivated.) # '''Predict''': What will happen if nothing is done? What could happen if X changed? What if Y or Z changed? Consider options. # '''Intervene''': Apply motivation and emotion principles. Implement a step-by-step plan to alter: #* environment #* relationships #* motivation #* emotion # '''Observe/measure/adjust''': #* Measure key processes and outcomes (before, during, after) #* Track progress regularly #* Adjust strategies based on evidence of what works or not #* Document what was done, what changes, and why Change is difficult. Complex problems often need persistence, feedback, and refinement. If an intervention doesn't work, analyse why, and try a revised approach. ==Experimental manipulations== <div align="center"> {| class="wikitable mw-collapsible" |+ Examples of brief experimental manipulations ! Ch ! Motivation/Emotion ! Experimental manipulation |- | 3 | Trust | Oxytocin hormone via nasal spray |- | 4 | Sexual attraction | Facial metrics (large eyes, small nose, small chin) |- | 5 | Extrinsic motivation | Offering a scholarship for high grades |- | 6 | Autonomy | Self-direction in pursuit of a personal goal |- | 7 | Discrepancy | A difficult, specific, self-concordant goal to strive for |- | 8 | Growth mindset | Story about how Einstein worked hard to become smart |- | 9 | Self-efficacy | A highly competent role model |- | 10 | Self-control depletion | Resisting an attractive temptation for 5 minutes |- | 11 | Positive affect | Receiving a small, unexpected gift |- | 12 | Sadness | Viewing a film about a son at his Dad’s funeral |- | 13 | Embarrassment | Committing a social blunder in front of an audience |- | 14 | Congruence | Unconditional positive regard |- | 15 | Obsession | Keep a secret about the person you are talking to |} </div> ''Note''. Based on Reeve (2024), Table 16.2, p. 350 ==Mini-theories== Use theory to help explain the cause of what is happening and possible forms of intervention. All theories are (potentially) "correct"—the practitioner's job is to match the moment (time, place, context) with a potentially useful mini-theory. Here's a list of some key mini-theories discussed in this unit: * Achievement motivation * Arousal * Attribution * Basic psychological needs * Broaden-and-build * Cognitive dissonance * Cognitive evaluation theory * Drive * Ego depletion * Ego development * Facial feedback * Flow * Fully functioning person * Goal setting * Intrinsic-extrinsic motivation * Learned helplessness * Mindsets * Positive affect * Psychodynamics * Reactance * Self-actualisation * Self-concordance * Self-determination * Self-efficacy * Stress and coping What theories could help predict motivational and/or emotional responses to the following conditions? * Deprivation from food and people * Seeing a threatening (angry) face * Smelling rotting food * Encountering an obstacle to one's goal * Expecting a tangible reward (e.g., money) * Uncontrollable environment * Failure in front of others * Observing a competent role model * Autonomy-supportive teacher ==Intervention strategies== There are two basic strategies for intervening: * '''Overcome''': Fix what isn't working ** Repair weaknesses ** Overcome pathology * '''Promote''' and further develop what is already working ** Amplify strengths ** Improve functioning The table below shows a motivational problem to be overcome on the left. On the right, is a theory-based strength to promote and develop. ==Practical problems== Whilst these scenarios lack context, it should still be possible to generate some ideas about: * Why? (explain) * What will happen; but also, what could be? (predict) * Implementation (application) of strategies Explain → Predict → Intervene: * Child not wanting to brush teeth * Sales employee quota * Elite athlete performance * Weight loss * Study motivation These scenarios are explained in more detail below (based on Reeve, 2018, Box 17, p. 426) ;Child at home A 6-year girl resists brushing her teeth every night before bed. She finds the routine boring and doesn’t understand why it’s important. When asked to brush her teeth, she plays with the water and brushes for only a few seconds. Her parents, aware of the long-term benefits of good dental hygiene, find themselves in a constant battle to get her to complete the task properly. None of their strategies seem to work. This nightly conflict drains the family’s energy and patience. ; Employee at work A mid-level sales representative at a large company feels increasingly disillusioned with her role. Despite knowing she has the skills to sell, she faces rejection on nearly every call. Her company only measures her success through monthly quotas, providing no support for personal development or job satisfaction. She meets her quota some months, but the constant rejection and lack of recognition make her feel drained. Her confidence has eroded, and she is beginning to question whether this job aligns with her strengths and interests. She fantasises about quitting but worries about financial stability. ; Athlete or musician A promising 18-year-old athlete (or musician) is in love with her craft. She spends hours practicing, enjoys the discipline, and consistently performs well. Despite her dedication, her growth in skill level has stagnated over the past year. While her peers are progressing rapidly, she feels stuck. Coaches and teachers praise her current ability but offer no new insights on how to break through to the next level. Frustrated, she worries that no matter how hard she works, she won't reach her goal of becoming an elite performer. Her motivation remains high, but she fears plateauing permanently. ;Medical patient A middle-aged man has been advised by his physician to lose 15 kilograms to avoid serious health risks, including a possible heart attack. He acknowledges the logic of the advice but feels disconnected from the urgency of the situation. A long history of poor dietary habits and a sedentary lifestyle has left him with little confidence that he can make meaningful changes. Despite multiple attempts to exercise and eat healthier in the past, nothing has ever stuck. He feels overwhelmed by the prospect of overhauling his lifestyle and is pessimistic that the results would justify the effort. ;Suffering student A university student in his third year is taking an advanced course in a subject he thought he could handle. Despite attending lectures and completing the readings, he performed poorly on the mid-term exam, far below his usual standards. The grade was a shock, leaving him feeling embarrassed and ashamed. He begins to doubt his abilities and contemplates giving up entirely. Each new lecture feels more overwhelming, and his efforts in studying feel futile. As the second half of the semester looms, his motivation plummets, and he wishes he could just drop the course. ==Nuggets of truth== [[File:Pearl 1.svg|right|50px]] [[File:Treasurchest.svg|right|50px]] After a tsunami of theories and research, let's gather your pearls or wisdom or nuggets of truth about the psychology motivation and emotion: * What are the '''greatest insights''', '''significant learnings''', or '''best ideas''' you've acquired through this unit? * What was the '''take-home message'''(s) from the chapter you created? Contribute your insights to: * [[Motivation and emotion/Nuggets of truth|Nuggets of truth]] (Wikiversity) <!-- Example book chapter take-home messages: * [https://x.com/jtneill/status/1451068316131287041 X] (2021) * [[Motivation and emotion/Book/2011/Pearls_of_wisdom|Wikiversity]] (2011) --> ==Wisdom gained== {{Nutshell| Motivation and emotion fuel human behaviour, shaped by unconscious drives and conscious goals, with social relations playing a key role. While we may underestimate the power of biological urges and emotional dynamics, intrinsic motivation and clear plans unlock our best performance. Self-efficacy and self-control help balance short-term impulses with long-term aspirations. The development of emotional intelligence guides us toward growth and flourishing. Understanding the functional complexity of motivation and emotion can help us to thrive as fully functionally people.}} Wisdom gained from a scientific study of motivation and emotion includes that: # '''Motivation''' = behavioural energy, direction, and persistence # '''What we don't know''' about motivation and emotion exceeds what we do know # The '''brain''' is as much about motivation and emotion as it is about cognition and thinking # We underestimate how powerful a motivational force '''biological urges''' can be when not experiencing them # '''Quality''' of motivation (intrinsic vs. extrinsic) is important # To flourish, motivation needs supportive conditions, especially '''supportive relationships''' # '''Implicit''' (unconscious) '''motives''' predict behaviour better than explicit (conscious) motives # We do our best when we have a specific plan of action to pursue a '''difficult, specific, and self-congruent goal''' # People with '''different mindsets''' pursue goals in different ways # The core '''self-efficacy beliefs''' of "I can do it" and "it will work" underlie competent functioning # Exert '''self-control over short-term urges''' to effectively pursue long-term goals # All '''emotions are good''' because they serve a functional purpose # '''Other people''' are the source of most of our emotions # The more sophisticated our '''emotional repertoire''', the more likely we are to have the right emotions in every situation # '''Encouraging growth''' is more productive than trying to cure weakness # Motivation often arises from outside of '''conscious awareness''' # There is nothing so practical as a '''good theory''' ==Readings== * Chapter 17: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 16: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) ==Slides== * [https://docs.google.com/presentation/d/1rpaBEkSs5p0TFT3i5JT_o0vG3wZkTJtFAdMO74h8teQ/edit?usp=sharing Interventions] (Google Slides) * [https://docs.google.com/presentation/d/1kucigYPF-72cP0FAbzQZmnOUhNfilMSsbGOQnrnWRaM/edit?usp=sharing Review] (Google Slides) <!-- * [http://www.slideshare.net/jtneill/conclusion-and-review Lecture slides] (Slideshare, 2016) * Handouts **[[Media:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|100px]] * 2014 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] * 2011 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ==Recordings== ;13/11/2013 {{Motivation and emotion/Lectures/Recording3|2013|5b4fdfbe-0073-498f-ad85-53d9c6d06e89}} ;16/11/2011 {{Motivation and emotion/Lectures/Recording|2011|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.m4v|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.mp3}} <!-- * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.mp3 mp3] (audio) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.m4v m4v] (video) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872 echo360] (rich media) --> ==See also== * [[/Images/]] ; Lecture * [[{{#titleparts:{{PAGENAME}}|2}}/Growth psychology|Growth psychology]] (Previous lecture) ; Tutorial * [[Motivation and emotion/Tutorials/Review|Review]] (Tutorial) ==Recording== * [https://au-lti.bbcollab.com/recording/b13748d85c474a87a061ab86fdc19adc Lecture 12] (2025)<!-- * [https://au-lti.bbcollab.com/recording/daa1ba979ba045799efbe73f3226b00f Lecture 12] (2024) * [https://au-lti.bbcollab.com/recording/8ef2ff0521b1457197d7920a93cffc99 Lecture 12] (2023) * [https://au-lti.bbcollab.com/recording/53b16231d9904444bac31b14bc7784f6 Lecture 12 recording] (2022) * [https://au-lti.bbcollab.com/recording/0bd459db9ece46b388a4b794f5751cff Lecture 12 recording] (2021) * [https://au-lti.bbcollab.com/recording/e810c154cfbe41189f8fb4056c871206 Lecture 12 recording] (2020) * [https://echo360.org.au/media/ea2a5707-6be8-43fc-996c-e176ce54bc4f/public Lecture 12 recording] (2018) * [https://echo360.org.au/media/2417e774-f589-4635-9955-9aaf73bcd3b8/public Lecture 12 recording] (2017)--> ==References== {{Hanging indent|1= Cheon, S. H., Reeve, J., & Moon, I. S. (2012). Experimentally based, longitudinally designed, teacher-focused intervention to help physical education teachers be more autonomy supportive toward their students. ''Journal of Sport and Exercise Psychology'', ''34''(3), 365–396. https://doi.org/10.1123/jsep.34.3.365 Izard, C. E., King, K. A., Trentacosta, C. J., Morgan, J. K., Laurenceau, J.-P., Krauthamer-Ewing, S. E., & Finlon, K. J. (2008). Accelerating the development of emotion competence in Head Start children: Effects on adaptive and maladaptive behavior. ''Development and Psychopathology'', ''20'', 369–397. https://doi.org/10.1017/S0954579408000175[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/82060B84CF144DBC4E1AC53CEC26A494/S0954579408000175a.pdf/accelerating_the_development_of_emotion_competence_in_head_start_children_effects_on_adaptive_and_maladaptive_behavior.pdf] Yeager, D. S., Trzesniewski, K. H., & Dweck, C. S. (2013). An implicit theory of personality intervention reduces adolescent aggression in response to victimization and exclusion. ''Child Development'', ''84'', 970–988. https://doi.org/10.1111/cdev.12003 }} <!-- ==External links== --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Interventions and review]] i7fko60l7pdgoph7jo7hvgi4gk70tnj 2844166 2844164 2026-11-03T10:44:09Z Jtneill 10242 /* Intervention strategies */ 2844166 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 12: Interventions and review|twelfth and final}} {{Motivation and emotion/Lectures/In development}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:Childcare Development Center-Crestwood High School cheerleaders 120815-F-PG936-400.jpg|right|250px|thumb|'''Figure 1'''. Interventions to change motivation and emotion are commonly used in educational settings.]] ==Overview== This lecture applies and reviews the [[motivation and emotion]] unit by discussing: * interventions using motivation and emotion principles * key points and wisdom gained Take-home messages: * Motivation guides engagement in behaviours that optimise well-being * Emotions provide feedback about progress towards goals * There is nothing so practical as a good theory ==Outline== * Interventions ** Interventions ** Practical problems ** Example interventions * Review ** Unit aims ** Nuggets of truth / Wisdom gained ** Review of key content ==Interventions== The ultimate test of a science is how well its principles can be applied to change desired outcomes. From a psychological point of view, motivation and emotion are dynamic and malleable and, therefore, can also be intentionally changed and strengthened. Well-designed interventions can develop motivational and emotional resources which help promote life outcomes that people care about (e.g., engagement, learning, performance, and well-being) (see Figure 1). The "trick" is not to try changing outcomes directly, but rather to ''alter the conditions'' so they are more conducive to the emergence of the desired outcomes. ==Plan== To systematically intervene, create and follow a plan: # '''Explain''': Diagnose–''Why'' is this happening? What is motivating this person? (Remember: ''All'' behaviour is motivated.) # '''Predict''': What will happen if nothing is done? What could happen if X changed? What if Y or Z changed? Consider options. # '''Intervene''': Apply motivation and emotion principles. Implement a step-by-step plan to alter: #* environment #* relationships #* motivation #* emotion # '''Observe/measure/adjust''': #* Measure key processes and outcomes (before, during, after) #* Track progress regularly #* Adjust strategies based on evidence of what works or not #* Document what was done, what changes, and why Change is difficult. Complex problems often need persistence, feedback, and refinement. If an intervention doesn't work, analyse why, and try a revised approach. ==Experimental manipulations== <div align="center"> {| class="wikitable mw-collapsible" |+ Examples of brief experimental manipulations ! Ch ! Motivation/Emotion ! Experimental manipulation |- | 3 | Trust | Oxytocin hormone via nasal spray |- | 4 | Sexual attraction | Facial metrics (large eyes, small nose, small chin) |- | 5 | Extrinsic motivation | Offering a scholarship for high grades |- | 6 | Autonomy | Self-direction in pursuit of a personal goal |- | 7 | Discrepancy | A difficult, specific, self-concordant goal to strive for |- | 8 | Growth mindset | Story about how Einstein worked hard to become smart |- | 9 | Self-efficacy | A highly competent role model |- | 10 | Self-control depletion | Resisting an attractive temptation for 5 minutes |- | 11 | Positive affect | Receiving a small, unexpected gift |- | 12 | Sadness | Viewing a film about a son at his Dad’s funeral |- | 13 | Embarrassment | Committing a social blunder in front of an audience |- | 14 | Congruence | Unconditional positive regard |- | 15 | Obsession | Keep a secret about the person you are talking to |} </div> ''Note''. Based on Reeve (2024), Table 16.2, p. 350 ==Mini-theories== Use theory to help explain the cause of what is happening and possible forms of intervention. All theories are (potentially) "correct"—the practitioner's job is to match the moment (time, place, context) with a potentially useful mini-theory. Here's a list of some key mini-theories discussed in this unit: * Achievement motivation * Arousal * Attribution * Basic psychological needs * Broaden-and-build * Cognitive dissonance * Cognitive evaluation theory * Drive * Ego depletion * Ego development * Facial feedback * Flow * Fully functioning person * Goal setting * Intrinsic-extrinsic motivation * Learned helplessness * Mindsets * Positive affect * Psychodynamics * Reactance * Self-actualisation * Self-concordance * Self-determination * Self-efficacy * Stress and coping What theories could help predict motivational and/or emotional responses to the following conditions? * Deprivation from food and people * Seeing a threatening (angry) face * Smelling rotting food * Encountering an obstacle to one's goal * Expecting a tangible reward (e.g., money) * Uncontrollable environment * Failure in front of others * Observing a competent role model * Autonomy-supportive teacher ==Intervention strategies== There are two basic strategies for intervening: * '''Overcome''': Fix what isn't working ** Repair weaknesses ** Overcome pathology * '''Promote''' and further develop what is already working ** Amplify strengths ** Improve functioning ==Practical problems== Whilst these scenarios lack context, it should still be possible to generate some ideas about: * Why? (explain) * What will happen; but also, what could be? (predict) * Implementation (application) of strategies Explain → Predict → Intervene: * Child not wanting to brush teeth * Sales employee quota * Elite athlete performance * Weight loss * Study motivation These scenarios are explained in more detail below (based on Reeve, 2018, Box 17, p. 426) ;Child at home A 6-year girl resists brushing her teeth every night before bed. She finds the routine boring and doesn’t understand why it’s important. When asked to brush her teeth, she plays with the water and brushes for only a few seconds. Her parents, aware of the long-term benefits of good dental hygiene, find themselves in a constant battle to get her to complete the task properly. None of their strategies seem to work. This nightly conflict drains the family’s energy and patience. ; Employee at work A mid-level sales representative at a large company feels increasingly disillusioned with her role. Despite knowing she has the skills to sell, she faces rejection on nearly every call. Her company only measures her success through monthly quotas, providing no support for personal development or job satisfaction. She meets her quota some months, but the constant rejection and lack of recognition make her feel drained. Her confidence has eroded, and she is beginning to question whether this job aligns with her strengths and interests. She fantasises about quitting but worries about financial stability. ; Athlete or musician A promising 18-year-old athlete (or musician) is in love with her craft. She spends hours practicing, enjoys the discipline, and consistently performs well. Despite her dedication, her growth in skill level has stagnated over the past year. While her peers are progressing rapidly, she feels stuck. Coaches and teachers praise her current ability but offer no new insights on how to break through to the next level. Frustrated, she worries that no matter how hard she works, she won't reach her goal of becoming an elite performer. Her motivation remains high, but she fears plateauing permanently. ;Medical patient A middle-aged man has been advised by his physician to lose 15 kilograms to avoid serious health risks, including a possible heart attack. He acknowledges the logic of the advice but feels disconnected from the urgency of the situation. A long history of poor dietary habits and a sedentary lifestyle has left him with little confidence that he can make meaningful changes. Despite multiple attempts to exercise and eat healthier in the past, nothing has ever stuck. He feels overwhelmed by the prospect of overhauling his lifestyle and is pessimistic that the results would justify the effort. ;Suffering student A university student in his third year is taking an advanced course in a subject he thought he could handle. Despite attending lectures and completing the readings, he performed poorly on the mid-term exam, far below his usual standards. The grade was a shock, leaving him feeling embarrassed and ashamed. He begins to doubt his abilities and contemplates giving up entirely. Each new lecture feels more overwhelming, and his efforts in studying feel futile. As the second half of the semester looms, his motivation plummets, and he wishes he could just drop the course. ==Nuggets of truth== [[File:Pearl 1.svg|right|50px]] [[File:Treasurchest.svg|right|50px]] After a tsunami of theories and research, let's gather your pearls or wisdom or nuggets of truth about the psychology motivation and emotion: * What are the '''greatest insights''', '''significant learnings''', or '''best ideas''' you've acquired through this unit? * What was the '''take-home message'''(s) from the chapter you created? Contribute your insights to: * [[Motivation and emotion/Nuggets of truth|Nuggets of truth]] (Wikiversity) <!-- Example book chapter take-home messages: * [https://x.com/jtneill/status/1451068316131287041 X] (2021) * [[Motivation and emotion/Book/2011/Pearls_of_wisdom|Wikiversity]] (2011) --> ==Wisdom gained== {{Nutshell| Motivation and emotion fuel human behaviour, shaped by unconscious drives and conscious goals, with social relations playing a key role. While we may underestimate the power of biological urges and emotional dynamics, intrinsic motivation and clear plans unlock our best performance. Self-efficacy and self-control help balance short-term impulses with long-term aspirations. The development of emotional intelligence guides us toward growth and flourishing. Understanding the functional complexity of motivation and emotion can help us to thrive as fully functionally people.}} Wisdom gained from a scientific study of motivation and emotion includes that: # '''Motivation''' = behavioural energy, direction, and persistence # '''What we don't know''' about motivation and emotion exceeds what we do know # The '''brain''' is as much about motivation and emotion as it is about cognition and thinking # We underestimate how powerful a motivational force '''biological urges''' can be when not experiencing them # '''Quality''' of motivation (intrinsic vs. extrinsic) is important # To flourish, motivation needs supportive conditions, especially '''supportive relationships''' # '''Implicit''' (unconscious) '''motives''' predict behaviour better than explicit (conscious) motives # We do our best when we have a specific plan of action to pursue a '''difficult, specific, and self-congruent goal''' # People with '''different mindsets''' pursue goals in different ways # The core '''self-efficacy beliefs''' of "I can do it" and "it will work" underlie competent functioning # Exert '''self-control over short-term urges''' to effectively pursue long-term goals # All '''emotions are good''' because they serve a functional purpose # '''Other people''' are the source of most of our emotions # The more sophisticated our '''emotional repertoire''', the more likely we are to have the right emotions in every situation # '''Encouraging growth''' is more productive than trying to cure weakness # Motivation often arises from outside of '''conscious awareness''' # There is nothing so practical as a '''good theory''' ==Readings== * Chapter 17: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 16: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) ==Slides== * [https://docs.google.com/presentation/d/1rpaBEkSs5p0TFT3i5JT_o0vG3wZkTJtFAdMO74h8teQ/edit?usp=sharing Interventions] (Google Slides) * [https://docs.google.com/presentation/d/1kucigYPF-72cP0FAbzQZmnOUhNfilMSsbGOQnrnWRaM/edit?usp=sharing Review] (Google Slides) <!-- * [http://www.slideshare.net/jtneill/conclusion-and-review Lecture slides] (Slideshare, 2016) * Handouts **[[Media:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|100px]] * 2014 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] * 2011 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ==Recordings== ;13/11/2013 {{Motivation and emotion/Lectures/Recording3|2013|5b4fdfbe-0073-498f-ad85-53d9c6d06e89}} ;16/11/2011 {{Motivation and emotion/Lectures/Recording|2011|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.m4v|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.mp3}} <!-- * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.mp3 mp3] (audio) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.m4v m4v] (video) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872 echo360] (rich media) --> ==See also== * [[/Images/]] ; Lecture * [[{{#titleparts:{{PAGENAME}}|2}}/Growth psychology|Growth psychology]] (Previous lecture) ; Tutorial * [[Motivation and emotion/Tutorials/Review|Review]] (Tutorial) ==Recording== * [https://au-lti.bbcollab.com/recording/b13748d85c474a87a061ab86fdc19adc Lecture 12] (2025)<!-- * [https://au-lti.bbcollab.com/recording/daa1ba979ba045799efbe73f3226b00f Lecture 12] (2024) * [https://au-lti.bbcollab.com/recording/8ef2ff0521b1457197d7920a93cffc99 Lecture 12] (2023) * [https://au-lti.bbcollab.com/recording/53b16231d9904444bac31b14bc7784f6 Lecture 12 recording] (2022) * [https://au-lti.bbcollab.com/recording/0bd459db9ece46b388a4b794f5751cff Lecture 12 recording] (2021) * [https://au-lti.bbcollab.com/recording/e810c154cfbe41189f8fb4056c871206 Lecture 12 recording] (2020) * [https://echo360.org.au/media/ea2a5707-6be8-43fc-996c-e176ce54bc4f/public Lecture 12 recording] (2018) * [https://echo360.org.au/media/2417e774-f589-4635-9955-9aaf73bcd3b8/public Lecture 12 recording] (2017)--> ==References== {{Hanging indent|1= Cheon, S. H., Reeve, J., & Moon, I. S. (2012). Experimentally based, longitudinally designed, teacher-focused intervention to help physical education teachers be more autonomy supportive toward their students. ''Journal of Sport and Exercise Psychology'', ''34''(3), 365–396. https://doi.org/10.1123/jsep.34.3.365 Izard, C. E., King, K. A., Trentacosta, C. J., Morgan, J. K., Laurenceau, J.-P., Krauthamer-Ewing, S. E., & Finlon, K. J. (2008). Accelerating the development of emotion competence in Head Start children: Effects on adaptive and maladaptive behavior. ''Development and Psychopathology'', ''20'', 369–397. https://doi.org/10.1017/S0954579408000175[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/82060B84CF144DBC4E1AC53CEC26A494/S0954579408000175a.pdf/accelerating_the_development_of_emotion_competence_in_head_start_children_effects_on_adaptive_and_maladaptive_behavior.pdf] Yeager, D. S., Trzesniewski, K. H., & Dweck, C. S. (2013). An implicit theory of personality intervention reduces adolescent aggression in response to victimization and exclusion. ''Child Development'', ''84'', 970–988. https://doi.org/10.1111/cdev.12003 }} <!-- ==External links== --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Interventions and review]] c2fovd4ur88w1xazpy54xbn1rllkewa 2844171 2844166 2026-11-03T10:51:02Z Jtneill 10242 /* Practical problems */ Revise and use feature boxes for scenarios 2844171 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 12: Interventions and review|twelfth and final}} {{Motivation and emotion/Lectures/In development}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:Childcare Development Center-Crestwood High School cheerleaders 120815-F-PG936-400.jpg|right|250px|thumb|'''Figure 1'''. Interventions to change motivation and emotion are commonly used in educational settings.]] ==Overview== This lecture applies and reviews the [[motivation and emotion]] unit by discussing: * interventions using motivation and emotion principles * key points and wisdom gained Take-home messages: * Motivation guides engagement in behaviours that optimise well-being * Emotions provide feedback about progress towards goals * There is nothing so practical as a good theory ==Outline== * Interventions ** Interventions ** Practical problems ** Example interventions * Review ** Unit aims ** Nuggets of truth / Wisdom gained ** Review of key content ==Interventions== The ultimate test of a science is how well its principles can be applied to change desired outcomes. From a psychological point of view, motivation and emotion are dynamic and malleable and, therefore, can also be intentionally changed and strengthened. Well-designed interventions can develop motivational and emotional resources which help promote life outcomes that people care about (e.g., engagement, learning, performance, and well-being) (see Figure 1). The "trick" is not to try changing outcomes directly, but rather to ''alter the conditions'' so they are more conducive to the emergence of the desired outcomes. ==Plan== To systematically intervene, create and follow a plan: # '''Explain''': Diagnose–''Why'' is this happening? What is motivating this person? (Remember: ''All'' behaviour is motivated.) # '''Predict''': What will happen if nothing is done? What could happen if X changed? What if Y or Z changed? Consider options. # '''Intervene''': Apply motivation and emotion principles. Implement a step-by-step plan to alter: #* environment #* relationships #* motivation #* emotion # '''Observe/measure/adjust''': #* Measure key processes and outcomes (before, during, after) #* Track progress regularly #* Adjust strategies based on evidence of what works or not #* Document what was done, what changes, and why Change is difficult. Complex problems often need persistence, feedback, and refinement. If an intervention doesn't work, analyse why, and try a revised approach. ==Experimental manipulations== <div align="center"> {| class="wikitable mw-collapsible" |+ Examples of brief experimental manipulations ! Ch ! Motivation/Emotion ! Experimental manipulation |- | 3 | Trust | Oxytocin hormone via nasal spray |- | 4 | Sexual attraction | Facial metrics (large eyes, small nose, small chin) |- | 5 | Extrinsic motivation | Offering a scholarship for high grades |- | 6 | Autonomy | Self-direction in pursuit of a personal goal |- | 7 | Discrepancy | A difficult, specific, self-concordant goal to strive for |- | 8 | Growth mindset | Story about how Einstein worked hard to become smart |- | 9 | Self-efficacy | A highly competent role model |- | 10 | Self-control depletion | Resisting an attractive temptation for 5 minutes |- | 11 | Positive affect | Receiving a small, unexpected gift |- | 12 | Sadness | Viewing a film about a son at his Dad’s funeral |- | 13 | Embarrassment | Committing a social blunder in front of an audience |- | 14 | Congruence | Unconditional positive regard |- | 15 | Obsession | Keep a secret about the person you are talking to |} </div> ''Note''. Based on Reeve (2024), Table 16.2, p. 350 ==Mini-theories== Use theory to help explain the cause of what is happening and possible forms of intervention. All theories are (potentially) "correct"—the practitioner's job is to match the moment (time, place, context) with a potentially useful mini-theory. Here's a list of some key mini-theories discussed in this unit: * Achievement motivation * Arousal * Attribution * Basic psychological needs * Broaden-and-build * Cognitive dissonance * Cognitive evaluation theory * Drive * Ego depletion * Ego development * Facial feedback * Flow * Fully functioning person * Goal setting * Intrinsic-extrinsic motivation * Learned helplessness * Mindsets * Positive affect * Psychodynamics * Reactance * Self-actualisation * Self-concordance * Self-determination * Self-efficacy * Stress and coping What theories could help predict motivational and/or emotional responses to the following conditions? * Deprivation from food and people * Seeing a threatening (angry) face * Smelling rotting food * Encountering an obstacle to one's goal * Expecting a tangible reward (e.g., money) * Uncontrollable environment * Failure in front of others * Observing a competent role model * Autonomy-supportive teacher ==Intervention strategies== There are two basic strategies for intervening: * '''Overcome''': Fix what isn't working ** Repair weaknesses ** Overcome pathology * '''Promote''' and further develop what is already working ** Amplify strengths ** Improve functioning ==Practical problems== What intervention ideas, based on motivation or emotion theory, can you generate for the following scenarios? * Child not wanting to brush teeth * Sales employee quota * Elite athlete performance * Weight loss * Study motivation These scenarios are explained in more detail below (based on Reeve, 2024, Box 16, p. 348) Guide: * Why? (Explain) * What will happen; what could be? (predict) (Predict) * Implementation (application) of strategies (Intervene) Explain → Predict → Intervene: {{RoundBoxTop|theme=3}} ;Child not wanting to brush teeth A 6-year girl resists brushing her teeth every night before bed. She finds the routine boring and doesn't understand why it's important. When asked to brush her teeth, she plays with the water and brushes for only a few seconds. Her parents, aware of the long-term benefits of good dental hygiene, find themselves in a constant battle to get her to complete the task properly. None of their strategies seem to work. This nightly conflict drains the family's energy and patience. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Sales employee quota A mid-level sales representative at a large company feels increasingly disillusioned with her role. Despite knowing she has the skills to sell, she faces rejection on nearly every call. Her company only measures her success through monthly quotas, providing no support for personal development or job satisfaction. She meets her quota some months, but the constant rejection and lack of recognition make her feel drained. Her confidence has eroded, and she is beginning to question whether this job aligns with her strengths and interests. She fantasises about quitting but worries about financial stability. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Elite athlete performance A promising 18-year-old athlete (or musician) is in love with her craft. She spends hours practicing, enjoys the discipline, and consistently performs well. Despite her dedication, her growth in skill level has stagnated over the past year. While her peers are progressing rapidly, she feels stuck. Coaches and teachers praise her current ability but offer no new insights on how to break through to the next level. Frustrated, she worries that no matter how hard she works, she won't reach her goal of becoming an elite performer. Her motivation remains high, but she fears plateauing permanently. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Weight loss A middle-aged man has been advised by his physician to lose 15 kilograms to avoid serious health risks, including a possible heart attack. He acknowledges the logic of the advice but feels disconnected from the urgency of the situation. A long history of poor dietary habits and a sedentary lifestyle has left him with little confidence that he can make meaningful changes. Despite multiple attempts to exercise and eat healthier in the past, nothing has ever stuck. He feels overwhelmed by the prospect of overhauling his lifestyle and is pessimistic that the results would justify the effort. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Study motivation A university student in his third year is taking an advanced course in a subject he thought he could handle. Despite attending lectures and completing the readings, he performed poorly on the mid-term exam, far below his usual standards. The grade was a shock, leaving him feeling embarrassed and ashamed. He begins to doubt his abilities and contemplates giving up entirely. Each new lecture feels more overwhelming, and his efforts in studying feel futile. As the second half of the semester looms, his motivation plummets, and he wishes he could just drop the course. {{RoundBoxBottom}} ==Nuggets of truth== [[File:Pearl 1.svg|right|50px]] [[File:Treasurchest.svg|right|50px]] After a tsunami of theories and research, let's gather your pearls or wisdom or nuggets of truth about the psychology motivation and emotion: * What are the '''greatest insights''', '''significant learnings''', or '''best ideas''' you've acquired through this unit? * What was the '''take-home message'''(s) from the chapter you created? Contribute your insights to: * [[Motivation and emotion/Nuggets of truth|Nuggets of truth]] (Wikiversity) <!-- Example book chapter take-home messages: * [https://x.com/jtneill/status/1451068316131287041 X] (2021) * [[Motivation and emotion/Book/2011/Pearls_of_wisdom|Wikiversity]] (2011) --> ==Wisdom gained== {{Nutshell| Motivation and emotion fuel human behaviour, shaped by unconscious drives and conscious goals, with social relations playing a key role. While we may underestimate the power of biological urges and emotional dynamics, intrinsic motivation and clear plans unlock our best performance. Self-efficacy and self-control help balance short-term impulses with long-term aspirations. The development of emotional intelligence guides us toward growth and flourishing. Understanding the functional complexity of motivation and emotion can help us to thrive as fully functionally people.}} Wisdom gained from a scientific study of motivation and emotion includes that: # '''Motivation''' = behavioural energy, direction, and persistence # '''What we don't know''' about motivation and emotion exceeds what we do know # The '''brain''' is as much about motivation and emotion as it is about cognition and thinking # We underestimate how powerful a motivational force '''biological urges''' can be when not experiencing them # '''Quality''' of motivation (intrinsic vs. extrinsic) is important # To flourish, motivation needs supportive conditions, especially '''supportive relationships''' # '''Implicit''' (unconscious) '''motives''' predict behaviour better than explicit (conscious) motives # We do our best when we have a specific plan of action to pursue a '''difficult, specific, and self-congruent goal''' # People with '''different mindsets''' pursue goals in different ways # The core '''self-efficacy beliefs''' of "I can do it" and "it will work" underlie competent functioning # Exert '''self-control over short-term urges''' to effectively pursue long-term goals # All '''emotions are good''' because they serve a functional purpose # '''Other people''' are the source of most of our emotions # The more sophisticated our '''emotional repertoire''', the more likely we are to have the right emotions in every situation # '''Encouraging growth''' is more productive than trying to cure weakness # Motivation often arises from outside of '''conscious awareness''' # There is nothing so practical as a '''good theory''' ==Readings== * Chapter 17: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 16: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) ==Slides== * [https://docs.google.com/presentation/d/1rpaBEkSs5p0TFT3i5JT_o0vG3wZkTJtFAdMO74h8teQ/edit?usp=sharing Interventions] (Google Slides) * [https://docs.google.com/presentation/d/1kucigYPF-72cP0FAbzQZmnOUhNfilMSsbGOQnrnWRaM/edit?usp=sharing Review] (Google Slides) <!-- * [http://www.slideshare.net/jtneill/conclusion-and-review Lecture slides] (Slideshare, 2016) * Handouts **[[Media:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|100px]] * 2014 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] * 2011 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ==Recordings== ;13/11/2013 {{Motivation and emotion/Lectures/Recording3|2013|5b4fdfbe-0073-498f-ad85-53d9c6d06e89}} ;16/11/2011 {{Motivation and emotion/Lectures/Recording|2011|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.m4v|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.mp3}} <!-- * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.mp3 mp3] (audio) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.m4v m4v] (video) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872 echo360] (rich media) --> ==See also== * [[/Images/]] ; Lecture * [[{{#titleparts:{{PAGENAME}}|2}}/Growth psychology|Growth psychology]] (Previous lecture) ; Tutorial * [[Motivation and emotion/Tutorials/Review|Review]] (Tutorial) ==Recording== * [https://au-lti.bbcollab.com/recording/b13748d85c474a87a061ab86fdc19adc Lecture 12] (2025)<!-- * [https://au-lti.bbcollab.com/recording/daa1ba979ba045799efbe73f3226b00f Lecture 12] (2024) * [https://au-lti.bbcollab.com/recording/8ef2ff0521b1457197d7920a93cffc99 Lecture 12] (2023) * [https://au-lti.bbcollab.com/recording/53b16231d9904444bac31b14bc7784f6 Lecture 12 recording] (2022) * [https://au-lti.bbcollab.com/recording/0bd459db9ece46b388a4b794f5751cff Lecture 12 recording] (2021) * [https://au-lti.bbcollab.com/recording/e810c154cfbe41189f8fb4056c871206 Lecture 12 recording] (2020) * [https://echo360.org.au/media/ea2a5707-6be8-43fc-996c-e176ce54bc4f/public Lecture 12 recording] (2018) * [https://echo360.org.au/media/2417e774-f589-4635-9955-9aaf73bcd3b8/public Lecture 12 recording] (2017)--> ==References== {{Hanging indent|1= Cheon, S. H., Reeve, J., & Moon, I. S. (2012). Experimentally based, longitudinally designed, teacher-focused intervention to help physical education teachers be more autonomy supportive toward their students. ''Journal of Sport and Exercise Psychology'', ''34''(3), 365–396. https://doi.org/10.1123/jsep.34.3.365 Izard, C. E., King, K. A., Trentacosta, C. J., Morgan, J. K., Laurenceau, J.-P., Krauthamer-Ewing, S. E., & Finlon, K. J. (2008). Accelerating the development of emotion competence in Head Start children: Effects on adaptive and maladaptive behavior. ''Development and Psychopathology'', ''20'', 369–397. https://doi.org/10.1017/S0954579408000175[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/82060B84CF144DBC4E1AC53CEC26A494/S0954579408000175a.pdf/accelerating_the_development_of_emotion_competence_in_head_start_children_effects_on_adaptive_and_maladaptive_behavior.pdf] Yeager, D. S., Trzesniewski, K. H., & Dweck, C. S. (2013). An implicit theory of personality intervention reduces adolescent aggression in response to victimization and exclusion. ''Child Development'', ''84'', 970–988. https://doi.org/10.1111/cdev.12003 }} <!-- ==External links== --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Interventions and review]] g1ifjunp5gy96rb3z7cabhllv94w2qk 2844172 2844171 2026-11-03T10:52:01Z Jtneill 10242 /* Wisdom gained */ 2844172 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 12: Interventions and review|twelfth and final}} {{Motivation and emotion/Lectures/In development}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:Childcare Development Center-Crestwood High School cheerleaders 120815-F-PG936-400.jpg|right|250px|thumb|'''Figure 1'''. Interventions to change motivation and emotion are commonly used in educational settings.]] ==Overview== This lecture applies and reviews the [[motivation and emotion]] unit by discussing: * interventions using motivation and emotion principles * key points and wisdom gained Take-home messages: * Motivation guides engagement in behaviours that optimise well-being * Emotions provide feedback about progress towards goals * There is nothing so practical as a good theory ==Outline== * Interventions ** Interventions ** Practical problems ** Example interventions * Review ** Unit aims ** Nuggets of truth / Wisdom gained ** Review of key content ==Interventions== The ultimate test of a science is how well its principles can be applied to change desired outcomes. From a psychological point of view, motivation and emotion are dynamic and malleable and, therefore, can also be intentionally changed and strengthened. Well-designed interventions can develop motivational and emotional resources which help promote life outcomes that people care about (e.g., engagement, learning, performance, and well-being) (see Figure 1). The "trick" is not to try changing outcomes directly, but rather to ''alter the conditions'' so they are more conducive to the emergence of the desired outcomes. ==Plan== To systematically intervene, create and follow a plan: # '''Explain''': Diagnose–''Why'' is this happening? What is motivating this person? (Remember: ''All'' behaviour is motivated.) # '''Predict''': What will happen if nothing is done? What could happen if X changed? What if Y or Z changed? Consider options. # '''Intervene''': Apply motivation and emotion principles. Implement a step-by-step plan to alter: #* environment #* relationships #* motivation #* emotion # '''Observe/measure/adjust''': #* Measure key processes and outcomes (before, during, after) #* Track progress regularly #* Adjust strategies based on evidence of what works or not #* Document what was done, what changes, and why Change is difficult. Complex problems often need persistence, feedback, and refinement. If an intervention doesn't work, analyse why, and try a revised approach. ==Experimental manipulations== <div align="center"> {| class="wikitable mw-collapsible" |+ Examples of brief experimental manipulations ! Ch ! Motivation/Emotion ! Experimental manipulation |- | 3 | Trust | Oxytocin hormone via nasal spray |- | 4 | Sexual attraction | Facial metrics (large eyes, small nose, small chin) |- | 5 | Extrinsic motivation | Offering a scholarship for high grades |- | 6 | Autonomy | Self-direction in pursuit of a personal goal |- | 7 | Discrepancy | A difficult, specific, self-concordant goal to strive for |- | 8 | Growth mindset | Story about how Einstein worked hard to become smart |- | 9 | Self-efficacy | A highly competent role model |- | 10 | Self-control depletion | Resisting an attractive temptation for 5 minutes |- | 11 | Positive affect | Receiving a small, unexpected gift |- | 12 | Sadness | Viewing a film about a son at his Dad’s funeral |- | 13 | Embarrassment | Committing a social blunder in front of an audience |- | 14 | Congruence | Unconditional positive regard |- | 15 | Obsession | Keep a secret about the person you are talking to |} </div> ''Note''. Based on Reeve (2024), Table 16.2, p. 350 ==Mini-theories== Use theory to help explain the cause of what is happening and possible forms of intervention. All theories are (potentially) "correct"—the practitioner's job is to match the moment (time, place, context) with a potentially useful mini-theory. Here's a list of some key mini-theories discussed in this unit: * Achievement motivation * Arousal * Attribution * Basic psychological needs * Broaden-and-build * Cognitive dissonance * Cognitive evaluation theory * Drive * Ego depletion * Ego development * Facial feedback * Flow * Fully functioning person * Goal setting * Intrinsic-extrinsic motivation * Learned helplessness * Mindsets * Positive affect * Psychodynamics * Reactance * Self-actualisation * Self-concordance * Self-determination * Self-efficacy * Stress and coping What theories could help predict motivational and/or emotional responses to the following conditions? * Deprivation from food and people * Seeing a threatening (angry) face * Smelling rotting food * Encountering an obstacle to one's goal * Expecting a tangible reward (e.g., money) * Uncontrollable environment * Failure in front of others * Observing a competent role model * Autonomy-supportive teacher ==Intervention strategies== There are two basic strategies for intervening: * '''Overcome''': Fix what isn't working ** Repair weaknesses ** Overcome pathology * '''Promote''' and further develop what is already working ** Amplify strengths ** Improve functioning ==Practical problems== What intervention ideas, based on motivation or emotion theory, can you generate for the following scenarios? * Child not wanting to brush teeth * Sales employee quota * Elite athlete performance * Weight loss * Study motivation These scenarios are explained in more detail below (based on Reeve, 2024, Box 16, p. 348) Guide: * Why? (Explain) * What will happen; what could be? (predict) (Predict) * Implementation (application) of strategies (Intervene) Explain → Predict → Intervene: {{RoundBoxTop|theme=3}} ;Child not wanting to brush teeth A 6-year girl resists brushing her teeth every night before bed. She finds the routine boring and doesn't understand why it's important. When asked to brush her teeth, she plays with the water and brushes for only a few seconds. Her parents, aware of the long-term benefits of good dental hygiene, find themselves in a constant battle to get her to complete the task properly. None of their strategies seem to work. This nightly conflict drains the family's energy and patience. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Sales employee quota A mid-level sales representative at a large company feels increasingly disillusioned with her role. Despite knowing she has the skills to sell, she faces rejection on nearly every call. Her company only measures her success through monthly quotas, providing no support for personal development or job satisfaction. She meets her quota some months, but the constant rejection and lack of recognition make her feel drained. Her confidence has eroded, and she is beginning to question whether this job aligns with her strengths and interests. She fantasises about quitting but worries about financial stability. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Elite athlete performance A promising 18-year-old athlete (or musician) is in love with her craft. She spends hours practicing, enjoys the discipline, and consistently performs well. Despite her dedication, her growth in skill level has stagnated over the past year. While her peers are progressing rapidly, she feels stuck. Coaches and teachers praise her current ability but offer no new insights on how to break through to the next level. Frustrated, she worries that no matter how hard she works, she won't reach her goal of becoming an elite performer. Her motivation remains high, but she fears plateauing permanently. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Weight loss A middle-aged man has been advised by his physician to lose 15 kilograms to avoid serious health risks, including a possible heart attack. He acknowledges the logic of the advice but feels disconnected from the urgency of the situation. A long history of poor dietary habits and a sedentary lifestyle has left him with little confidence that he can make meaningful changes. Despite multiple attempts to exercise and eat healthier in the past, nothing has ever stuck. He feels overwhelmed by the prospect of overhauling his lifestyle and is pessimistic that the results would justify the effort. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Study motivation A university student in his third year is taking an advanced course in a subject he thought he could handle. Despite attending lectures and completing the readings, he performed poorly on the mid-term exam, far below his usual standards. The grade was a shock, leaving him feeling embarrassed and ashamed. He begins to doubt his abilities and contemplates giving up entirely. Each new lecture feels more overwhelming, and his efforts in studying feel futile. As the second half of the semester looms, his motivation plummets, and he wishes he could just drop the course. {{RoundBoxBottom}} ==Nuggets of truth== [[File:Pearl 1.svg|right|50px]] [[File:Treasurchest.svg|right|50px]] After a tsunami of theories and research, let's gather your pearls or wisdom or nuggets of truth about the psychology motivation and emotion: * What are the '''greatest insights''', '''significant learnings''', or '''best ideas''' you've acquired through this unit? * What was the '''take-home message'''(s) from the chapter you created? Contribute your insights to: * [[Motivation and emotion/Nuggets of truth|Nuggets of truth]] (Wikiversity) <!-- Example book chapter take-home messages: * [https://x.com/jtneill/status/1451068316131287041 X] (2021) * [[Motivation and emotion/Book/2011/Pearls_of_wisdom|Wikiversity]] (2011) --> ==Wisdom gained== {{Nutshell| Motivation and emotion fuel human behaviour, shaped by unconscious drives and conscious goals, with social relations playing a key role. While we may underestimate the power of biological urges and emotional dynamics, intrinsic motivation and clear plans unlock our best performance. Self-efficacy and self-control help balance short-term impulses with long-term aspirations. The development of emotional intelligence guides us toward growth and flourishing. Understanding the functional complexity of motivation and emotion can help us to thrive as fully functionally people.}} Wisdom gained from a scientific study of motivation and emotion includes that: # '''Motivation''' = behavioural energy, direction, and persistence # '''What we don't know''' about motivation and emotion exceeds what we do know # The '''brain''' is as much about motivation and emotion as it is about cognition and thinking # We underestimate how powerful a motivational force '''biological urges''' can be when not experiencing them # '''Quality''' of motivation (intrinsic vs. extrinsic) is important # To flourish, motivation needs supportive conditions, especially '''supportive relationships''' <!-- # '''Implicit''' (unconscious) '''motives''' predict behaviour better than explicit (conscious) motives --> # We do our best when we have a specific plan of action to pursue a '''difficult, specific, and self-congruent goal''' # People with '''different mindsets''' pursue goals in different ways # The core '''self-efficacy beliefs''' of "I can do it" and "it will work" underlie competent functioning # Exert '''self-control over short-term urges''' to effectively pursue long-term goals # All '''emotions are good''' because they serve a functional purpose # '''Other people''' are the source of most of our emotions # The more sophisticated our '''emotional repertoire''', the more likely we are to have the right emotions in every situation # '''Encouraging growth''' is more productive than trying to cure weakness # Motivation often arises from outside of '''conscious awareness''' # There is nothing so practical as a '''good theory''' ==Readings== * Chapter 17: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 16: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) ==Slides== * [https://docs.google.com/presentation/d/1rpaBEkSs5p0TFT3i5JT_o0vG3wZkTJtFAdMO74h8teQ/edit?usp=sharing Interventions] (Google Slides) * [https://docs.google.com/presentation/d/1kucigYPF-72cP0FAbzQZmnOUhNfilMSsbGOQnrnWRaM/edit?usp=sharing Review] (Google Slides) <!-- * [http://www.slideshare.net/jtneill/conclusion-and-review Lecture slides] (Slideshare, 2016) * Handouts **[[Media:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|100px]] * 2014 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] * 2011 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ==Recordings== ;13/11/2013 {{Motivation and emotion/Lectures/Recording3|2013|5b4fdfbe-0073-498f-ad85-53d9c6d06e89}} ;16/11/2011 {{Motivation and emotion/Lectures/Recording|2011|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.m4v|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.mp3}} <!-- * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.mp3 mp3] (audio) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.m4v m4v] (video) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872 echo360] (rich media) --> ==See also== * [[/Images/]] ; Lecture * [[{{#titleparts:{{PAGENAME}}|2}}/Growth psychology|Growth psychology]] (Previous lecture) ; Tutorial * [[Motivation and emotion/Tutorials/Review|Review]] (Tutorial) ==Recording== * [https://au-lti.bbcollab.com/recording/b13748d85c474a87a061ab86fdc19adc Lecture 12] (2025)<!-- * [https://au-lti.bbcollab.com/recording/daa1ba979ba045799efbe73f3226b00f Lecture 12] (2024) * [https://au-lti.bbcollab.com/recording/8ef2ff0521b1457197d7920a93cffc99 Lecture 12] (2023) * [https://au-lti.bbcollab.com/recording/53b16231d9904444bac31b14bc7784f6 Lecture 12 recording] (2022) * [https://au-lti.bbcollab.com/recording/0bd459db9ece46b388a4b794f5751cff Lecture 12 recording] (2021) * [https://au-lti.bbcollab.com/recording/e810c154cfbe41189f8fb4056c871206 Lecture 12 recording] (2020) * [https://echo360.org.au/media/ea2a5707-6be8-43fc-996c-e176ce54bc4f/public Lecture 12 recording] (2018) * [https://echo360.org.au/media/2417e774-f589-4635-9955-9aaf73bcd3b8/public Lecture 12 recording] (2017)--> ==References== {{Hanging indent|1= Cheon, S. H., Reeve, J., & Moon, I. S. (2012). Experimentally based, longitudinally designed, teacher-focused intervention to help physical education teachers be more autonomy supportive toward their students. ''Journal of Sport and Exercise Psychology'', ''34''(3), 365–396. https://doi.org/10.1123/jsep.34.3.365 Izard, C. E., King, K. A., Trentacosta, C. J., Morgan, J. K., Laurenceau, J.-P., Krauthamer-Ewing, S. E., & Finlon, K. J. (2008). Accelerating the development of emotion competence in Head Start children: Effects on adaptive and maladaptive behavior. ''Development and Psychopathology'', ''20'', 369–397. https://doi.org/10.1017/S0954579408000175[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/82060B84CF144DBC4E1AC53CEC26A494/S0954579408000175a.pdf/accelerating_the_development_of_emotion_competence_in_head_start_children_effects_on_adaptive_and_maladaptive_behavior.pdf] Yeager, D. S., Trzesniewski, K. H., & Dweck, C. S. (2013). An implicit theory of personality intervention reduces adolescent aggression in response to victimization and exclusion. ''Child Development'', ''84'', 970–988. https://doi.org/10.1111/cdev.12003 }} <!-- ==External links== --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Interventions and review]] 3ivij3gc1gy2t85ld2b41linhy8wdwk 2844174 2844172 2026-11-03T10:57:06Z Jtneill 10242 /* Wisdom gained */ Update 2844174 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 12: Interventions and review|twelfth and final}} {{Motivation and emotion/Lectures/In development}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:Childcare Development Center-Crestwood High School cheerleaders 120815-F-PG936-400.jpg|right|250px|thumb|'''Figure 1'''. Interventions to change motivation and emotion are commonly used in educational settings.]] ==Overview== This lecture applies and reviews the [[motivation and emotion]] unit by discussing: * interventions using motivation and emotion principles * key points and wisdom gained Take-home messages: * Motivation guides engagement in behaviours that optimise well-being * Emotions provide feedback about progress towards goals * There is nothing so practical as a good theory ==Outline== * Interventions ** Interventions ** Practical problems ** Example interventions * Review ** Unit aims ** Nuggets of truth / Wisdom gained ** Review of key content ==Interventions== The ultimate test of a science is how well its principles can be applied to change desired outcomes. From a psychological point of view, motivation and emotion are dynamic and malleable and, therefore, can also be intentionally changed and strengthened. Well-designed interventions can develop motivational and emotional resources which help promote life outcomes that people care about (e.g., engagement, learning, performance, and well-being) (see Figure 1). The "trick" is not to try changing outcomes directly, but rather to ''alter the conditions'' so they are more conducive to the emergence of the desired outcomes. ==Plan== To systematically intervene, create and follow a plan: # '''Explain''': Diagnose–''Why'' is this happening? What is motivating this person? (Remember: ''All'' behaviour is motivated.) # '''Predict''': What will happen if nothing is done? What could happen if X changed? What if Y or Z changed? Consider options. # '''Intervene''': Apply motivation and emotion principles. Implement a step-by-step plan to alter: #* environment #* relationships #* motivation #* emotion # '''Observe/measure/adjust''': #* Measure key processes and outcomes (before, during, after) #* Track progress regularly #* Adjust strategies based on evidence of what works or not #* Document what was done, what changes, and why Change is difficult. Complex problems often need persistence, feedback, and refinement. If an intervention doesn't work, analyse why, and try a revised approach. ==Experimental manipulations== <div align="center"> {| class="wikitable mw-collapsible" |+ Examples of brief experimental manipulations ! Ch ! Motivation/Emotion ! Experimental manipulation |- | 3 | Trust | Oxytocin hormone via nasal spray |- | 4 | Sexual attraction | Facial metrics (large eyes, small nose, small chin) |- | 5 | Extrinsic motivation | Offering a scholarship for high grades |- | 6 | Autonomy | Self-direction in pursuit of a personal goal |- | 7 | Discrepancy | A difficult, specific, self-concordant goal to strive for |- | 8 | Growth mindset | Story about how Einstein worked hard to become smart |- | 9 | Self-efficacy | A highly competent role model |- | 10 | Self-control depletion | Resisting an attractive temptation for 5 minutes |- | 11 | Positive affect | Receiving a small, unexpected gift |- | 12 | Sadness | Viewing a film about a son at his Dad’s funeral |- | 13 | Embarrassment | Committing a social blunder in front of an audience |- | 14 | Congruence | Unconditional positive regard |- | 15 | Obsession | Keep a secret about the person you are talking to |} </div> ''Note''. Based on Reeve (2024), Table 16.2, p. 350 ==Mini-theories== Use theory to help explain the cause of what is happening and possible forms of intervention. All theories are (potentially) "correct"—the practitioner's job is to match the moment (time, place, context) with a potentially useful mini-theory. Here's a list of some key mini-theories discussed in this unit: * Achievement motivation * Arousal * Attribution * Basic psychological needs * Broaden-and-build * Cognitive dissonance * Cognitive evaluation theory * Drive * Ego depletion * Ego development * Facial feedback * Flow * Fully functioning person * Goal setting * Intrinsic-extrinsic motivation * Learned helplessness * Mindsets * Positive affect * Psychodynamics * Reactance * Self-actualisation * Self-concordance * Self-determination * Self-efficacy * Stress and coping What theories could help predict motivational and/or emotional responses to the following conditions? * Deprivation from food and people * Seeing a threatening (angry) face * Smelling rotting food * Encountering an obstacle to one's goal * Expecting a tangible reward (e.g., money) * Uncontrollable environment * Failure in front of others * Observing a competent role model * Autonomy-supportive teacher ==Intervention strategies== There are two basic strategies for intervening: * '''Overcome''': Fix what isn't working ** Repair weaknesses ** Overcome pathology * '''Promote''' and further develop what is already working ** Amplify strengths ** Improve functioning ==Practical problems== What intervention ideas, based on motivation or emotion theory, can you generate for the following scenarios? * Child not wanting to brush teeth * Sales employee quota * Elite athlete performance * Weight loss * Study motivation These scenarios are explained in more detail below (based on Reeve, 2024, Box 16, p. 348) Guide: * Why? (Explain) * What will happen; what could be? (predict) (Predict) * Implementation (application) of strategies (Intervene) Explain → Predict → Intervene: {{RoundBoxTop|theme=3}} ;Child not wanting to brush teeth A 6-year girl resists brushing her teeth every night before bed. She finds the routine boring and doesn't understand why it's important. When asked to brush her teeth, she plays with the water and brushes for only a few seconds. Her parents, aware of the long-term benefits of good dental hygiene, find themselves in a constant battle to get her to complete the task properly. None of their strategies seem to work. This nightly conflict drains the family's energy and patience. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Sales employee quota A mid-level sales representative at a large company feels increasingly disillusioned with her role. Despite knowing she has the skills to sell, she faces rejection on nearly every call. Her company only measures her success through monthly quotas, providing no support for personal development or job satisfaction. She meets her quota some months, but the constant rejection and lack of recognition make her feel drained. Her confidence has eroded, and she is beginning to question whether this job aligns with her strengths and interests. She fantasises about quitting but worries about financial stability. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Elite athlete performance A promising 18-year-old athlete (or musician) is in love with her craft. She spends hours practicing, enjoys the discipline, and consistently performs well. Despite her dedication, her growth in skill level has stagnated over the past year. While her peers are progressing rapidly, she feels stuck. Coaches and teachers praise her current ability but offer no new insights on how to break through to the next level. Frustrated, she worries that no matter how hard she works, she won't reach her goal of becoming an elite performer. Her motivation remains high, but she fears plateauing permanently. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Weight loss A middle-aged man has been advised by his physician to lose 15 kilograms to avoid serious health risks, including a possible heart attack. He acknowledges the logic of the advice but feels disconnected from the urgency of the situation. A long history of poor dietary habits and a sedentary lifestyle has left him with little confidence that he can make meaningful changes. Despite multiple attempts to exercise and eat healthier in the past, nothing has ever stuck. He feels overwhelmed by the prospect of overhauling his lifestyle and is pessimistic that the results would justify the effort. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Study motivation A university student in his third year is taking an advanced course in a subject he thought he could handle. Despite attending lectures and completing the readings, he performed poorly on the mid-term exam, far below his usual standards. The grade was a shock, leaving him feeling embarrassed and ashamed. He begins to doubt his abilities and contemplates giving up entirely. Each new lecture feels more overwhelming, and his efforts in studying feel futile. As the second half of the semester looms, his motivation plummets, and he wishes he could just drop the course. {{RoundBoxBottom}} ==Nuggets of truth== [[File:Pearl 1.svg|right|50px]] [[File:Treasurchest.svg|right|50px]] After a tsunami of theories and research, let's gather your pearls or wisdom or nuggets of truth about the psychology motivation and emotion: * What are the '''greatest insights''', '''significant learnings''', or '''best ideas''' you've acquired through this unit? * What was the '''take-home message'''(s) from the chapter you created? Contribute your insights to: * [[Motivation and emotion/Nuggets of truth|Nuggets of truth]] (Wikiversity) <!-- Example book chapter take-home messages: * [https://x.com/jtneill/status/1451068316131287041 X] (2021) * [[Motivation and emotion/Book/2011/Pearls_of_wisdom|Wikiversity]] (2011) --> ==Wisdom gained== {{Nutshell| Motivation and emotion fuel human behaviour, shaped by unconscious drives and conscious goals, with social relations playing a key role. While we may underestimate the power of biological urges and emotional dynamics, intrinsic motivation and clear plans unlock our best performance. Self-efficacy and self-control help balance short-term impulses with long-term aspirations. The development of emotional intelligence helps with emotion regulation. Expressing and pursing higher needs guides us towards growth, flourishing, and fulfilling our potential. Understanding the functional complexity of motivation and emotion can help optimise human productivity and well-being.}} Wisdom gained from a scientific study of motivation and emotion includes that: # '''Motivation''' = behavioural energy, direction, and persistence # '''What we don't know''' about motivation and emotion exceeds what we do know # The '''brain''' is as much about motivation and emotion as it is about cognition and thinking # We underestimate how powerful a motivational force '''biological urges''' can be when not experiencing them # '''Quality''' of motivation (intrinsic vs. extrinsic) is important # To flourish, motivation needs supportive conditions, especially '''supportive relationships''' <!-- # '''Implicit''' (unconscious) '''motives''' predict behaviour better than explicit (conscious) motives --> # We do our best when we have a specific plan of action to pursue a '''difficult, specific, and self-congruent goal''' # People with '''different mindsets''' pursue goals in different ways # The core '''self-efficacy beliefs''' of "I can do it" and "it will work" underlie competent functioning # Exert '''self-control over short-term urges''' to effectively pursue long-term goals # All '''emotions are good''' because they serve a functional purpose # '''Other people''' are the source of most of our emotions # The more sophisticated our '''emotional repertoire''', the more likely we are to have the right emotions in every situation # '''Encouraging growth''' is more productive than trying to cure weakness # Motivation often arises from outside of '''conscious awareness''' # There is nothing so practical as a '''good theory''' ==Readings== * Chapter 17: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve, 2018]]) or Chapter 16: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) ==Slides== * [https://docs.google.com/presentation/d/1rpaBEkSs5p0TFT3i5JT_o0vG3wZkTJtFAdMO74h8teQ/edit?usp=sharing Interventions] (Google Slides) * [https://docs.google.com/presentation/d/1kucigYPF-72cP0FAbzQZmnOUhNfilMSsbGOQnrnWRaM/edit?usp=sharing Review] (Google Slides) <!-- * [http://www.slideshare.net/jtneill/conclusion-and-review Lecture slides] (Slideshare, 2016) * Handouts **[[Media:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|100px]] * 2014 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] * 2011 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ==Recordings== ;13/11/2013 {{Motivation and emotion/Lectures/Recording3|2013|5b4fdfbe-0073-498f-ad85-53d9c6d06e89}} ;16/11/2011 {{Motivation and emotion/Lectures/Recording|2011|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.m4v|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.mp3}} <!-- * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.mp3 mp3] (audio) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.m4v m4v] (video) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872 echo360] (rich media) --> ==See also== * [[/Images/]] ; Lecture * [[{{#titleparts:{{PAGENAME}}|2}}/Growth psychology|Growth psychology]] (Previous lecture) ; Tutorial * [[Motivation and emotion/Tutorials/Review|Review]] (Tutorial) ==Recording== * [https://au-lti.bbcollab.com/recording/b13748d85c474a87a061ab86fdc19adc Lecture 12] (2025)<!-- * [https://au-lti.bbcollab.com/recording/daa1ba979ba045799efbe73f3226b00f Lecture 12] (2024) * [https://au-lti.bbcollab.com/recording/8ef2ff0521b1457197d7920a93cffc99 Lecture 12] (2023) * [https://au-lti.bbcollab.com/recording/53b16231d9904444bac31b14bc7784f6 Lecture 12 recording] (2022) * [https://au-lti.bbcollab.com/recording/0bd459db9ece46b388a4b794f5751cff Lecture 12 recording] (2021) * [https://au-lti.bbcollab.com/recording/e810c154cfbe41189f8fb4056c871206 Lecture 12 recording] (2020) * [https://echo360.org.au/media/ea2a5707-6be8-43fc-996c-e176ce54bc4f/public Lecture 12 recording] (2018) * [https://echo360.org.au/media/2417e774-f589-4635-9955-9aaf73bcd3b8/public Lecture 12 recording] (2017)--> ==References== {{Hanging indent|1= Cheon, S. H., Reeve, J., & Moon, I. S. (2012). Experimentally based, longitudinally designed, teacher-focused intervention to help physical education teachers be more autonomy supportive toward their students. ''Journal of Sport and Exercise Psychology'', ''34''(3), 365–396. https://doi.org/10.1123/jsep.34.3.365 Izard, C. E., King, K. A., Trentacosta, C. J., Morgan, J. K., Laurenceau, J.-P., Krauthamer-Ewing, S. E., & Finlon, K. J. (2008). Accelerating the development of emotion competence in Head Start children: Effects on adaptive and maladaptive behavior. ''Development and Psychopathology'', ''20'', 369–397. https://doi.org/10.1017/S0954579408000175[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/82060B84CF144DBC4E1AC53CEC26A494/S0954579408000175a.pdf/accelerating_the_development_of_emotion_competence_in_head_start_children_effects_on_adaptive_and_maladaptive_behavior.pdf] Yeager, D. S., Trzesniewski, K. H., & Dweck, C. S. (2013). An implicit theory of personality intervention reduces adolescent aggression in response to victimization and exclusion. ''Child Development'', ''84'', 970–988. https://doi.org/10.1111/cdev.12003 }} <!-- ==External links== --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Interventions and review]] 4kfqg441lojtyy91exacc64av8s88p9 2844175 2844174 2026-11-03T10:57:32Z Jtneill 10242 /* Readings */ Update for 2026 2844175 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 12: Interventions and review|twelfth and final}} {{Motivation and emotion/Lectures/In development}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:Childcare Development Center-Crestwood High School cheerleaders 120815-F-PG936-400.jpg|right|250px|thumb|'''Figure 1'''. Interventions to change motivation and emotion are commonly used in educational settings.]] ==Overview== This lecture applies and reviews the [[motivation and emotion]] unit by discussing: * interventions using motivation and emotion principles * key points and wisdom gained Take-home messages: * Motivation guides engagement in behaviours that optimise well-being * Emotions provide feedback about progress towards goals * There is nothing so practical as a good theory ==Outline== * Interventions ** Interventions ** Practical problems ** Example interventions * Review ** Unit aims ** Nuggets of truth / Wisdom gained ** Review of key content ==Interventions== The ultimate test of a science is how well its principles can be applied to change desired outcomes. From a psychological point of view, motivation and emotion are dynamic and malleable and, therefore, can also be intentionally changed and strengthened. Well-designed interventions can develop motivational and emotional resources which help promote life outcomes that people care about (e.g., engagement, learning, performance, and well-being) (see Figure 1). The "trick" is not to try changing outcomes directly, but rather to ''alter the conditions'' so they are more conducive to the emergence of the desired outcomes. ==Plan== To systematically intervene, create and follow a plan: # '''Explain''': Diagnose–''Why'' is this happening? What is motivating this person? (Remember: ''All'' behaviour is motivated.) # '''Predict''': What will happen if nothing is done? What could happen if X changed? What if Y or Z changed? Consider options. # '''Intervene''': Apply motivation and emotion principles. Implement a step-by-step plan to alter: #* environment #* relationships #* motivation #* emotion # '''Observe/measure/adjust''': #* Measure key processes and outcomes (before, during, after) #* Track progress regularly #* Adjust strategies based on evidence of what works or not #* Document what was done, what changes, and why Change is difficult. Complex problems often need persistence, feedback, and refinement. If an intervention doesn't work, analyse why, and try a revised approach. ==Experimental manipulations== <div align="center"> {| class="wikitable mw-collapsible" |+ Examples of brief experimental manipulations ! Ch ! Motivation/Emotion ! Experimental manipulation |- | 3 | Trust | Oxytocin hormone via nasal spray |- | 4 | Sexual attraction | Facial metrics (large eyes, small nose, small chin) |- | 5 | Extrinsic motivation | Offering a scholarship for high grades |- | 6 | Autonomy | Self-direction in pursuit of a personal goal |- | 7 | Discrepancy | A difficult, specific, self-concordant goal to strive for |- | 8 | Growth mindset | Story about how Einstein worked hard to become smart |- | 9 | Self-efficacy | A highly competent role model |- | 10 | Self-control depletion | Resisting an attractive temptation for 5 minutes |- | 11 | Positive affect | Receiving a small, unexpected gift |- | 12 | Sadness | Viewing a film about a son at his Dad’s funeral |- | 13 | Embarrassment | Committing a social blunder in front of an audience |- | 14 | Congruence | Unconditional positive regard |- | 15 | Obsession | Keep a secret about the person you are talking to |} </div> ''Note''. Based on Reeve (2024), Table 16.2, p. 350 ==Mini-theories== Use theory to help explain the cause of what is happening and possible forms of intervention. All theories are (potentially) "correct"—the practitioner's job is to match the moment (time, place, context) with a potentially useful mini-theory. Here's a list of some key mini-theories discussed in this unit: * Achievement motivation * Arousal * Attribution * Basic psychological needs * Broaden-and-build * Cognitive dissonance * Cognitive evaluation theory * Drive * Ego depletion * Ego development * Facial feedback * Flow * Fully functioning person * Goal setting * Intrinsic-extrinsic motivation * Learned helplessness * Mindsets * Positive affect * Psychodynamics * Reactance * Self-actualisation * Self-concordance * Self-determination * Self-efficacy * Stress and coping What theories could help predict motivational and/or emotional responses to the following conditions? * Deprivation from food and people * Seeing a threatening (angry) face * Smelling rotting food * Encountering an obstacle to one's goal * Expecting a tangible reward (e.g., money) * Uncontrollable environment * Failure in front of others * Observing a competent role model * Autonomy-supportive teacher ==Intervention strategies== There are two basic strategies for intervening: * '''Overcome''': Fix what isn't working ** Repair weaknesses ** Overcome pathology * '''Promote''' and further develop what is already working ** Amplify strengths ** Improve functioning ==Practical problems== What intervention ideas, based on motivation or emotion theory, can you generate for the following scenarios? * Child not wanting to brush teeth * Sales employee quota * Elite athlete performance * Weight loss * Study motivation These scenarios are explained in more detail below (based on Reeve, 2024, Box 16, p. 348) Guide: * Why? (Explain) * What will happen; what could be? (predict) (Predict) * Implementation (application) of strategies (Intervene) Explain → Predict → Intervene: {{RoundBoxTop|theme=3}} ;Child not wanting to brush teeth A 6-year girl resists brushing her teeth every night before bed. She finds the routine boring and doesn't understand why it's important. When asked to brush her teeth, she plays with the water and brushes for only a few seconds. Her parents, aware of the long-term benefits of good dental hygiene, find themselves in a constant battle to get her to complete the task properly. None of their strategies seem to work. This nightly conflict drains the family's energy and patience. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Sales employee quota A mid-level sales representative at a large company feels increasingly disillusioned with her role. Despite knowing she has the skills to sell, she faces rejection on nearly every call. Her company only measures her success through monthly quotas, providing no support for personal development or job satisfaction. She meets her quota some months, but the constant rejection and lack of recognition make her feel drained. Her confidence has eroded, and she is beginning to question whether this job aligns with her strengths and interests. She fantasises about quitting but worries about financial stability. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Elite athlete performance A promising 18-year-old athlete (or musician) is in love with her craft. She spends hours practicing, enjoys the discipline, and consistently performs well. Despite her dedication, her growth in skill level has stagnated over the past year. While her peers are progressing rapidly, she feels stuck. Coaches and teachers praise her current ability but offer no new insights on how to break through to the next level. Frustrated, she worries that no matter how hard she works, she won't reach her goal of becoming an elite performer. Her motivation remains high, but she fears plateauing permanently. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Weight loss A middle-aged man has been advised by his physician to lose 15 kilograms to avoid serious health risks, including a possible heart attack. He acknowledges the logic of the advice but feels disconnected from the urgency of the situation. A long history of poor dietary habits and a sedentary lifestyle has left him with little confidence that he can make meaningful changes. Despite multiple attempts to exercise and eat healthier in the past, nothing has ever stuck. He feels overwhelmed by the prospect of overhauling his lifestyle and is pessimistic that the results would justify the effort. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Study motivation A university student in his third year is taking an advanced course in a subject he thought he could handle. Despite attending lectures and completing the readings, he performed poorly on the mid-term exam, far below his usual standards. The grade was a shock, leaving him feeling embarrassed and ashamed. He begins to doubt his abilities and contemplates giving up entirely. Each new lecture feels more overwhelming, and his efforts in studying feel futile. As the second half of the semester looms, his motivation plummets, and he wishes he could just drop the course. {{RoundBoxBottom}} ==Nuggets of truth== [[File:Pearl 1.svg|right|50px]] [[File:Treasurchest.svg|right|50px]] After a tsunami of theories and research, let's gather your pearls or wisdom or nuggets of truth about the psychology motivation and emotion: * What are the '''greatest insights''', '''significant learnings''', or '''best ideas''' you've acquired through this unit? * What was the '''take-home message'''(s) from the chapter you created? Contribute your insights to: * [[Motivation and emotion/Nuggets of truth|Nuggets of truth]] (Wikiversity) <!-- Example book chapter take-home messages: * [https://x.com/jtneill/status/1451068316131287041 X] (2021) * [[Motivation and emotion/Book/2011/Pearls_of_wisdom|Wikiversity]] (2011) --> ==Wisdom gained== {{Nutshell| Motivation and emotion fuel human behaviour, shaped by unconscious drives and conscious goals, with social relations playing a key role. While we may underestimate the power of biological urges and emotional dynamics, intrinsic motivation and clear plans unlock our best performance. Self-efficacy and self-control help balance short-term impulses with long-term aspirations. The development of emotional intelligence helps with emotion regulation. Expressing and pursing higher needs guides us towards growth, flourishing, and fulfilling our potential. Understanding the functional complexity of motivation and emotion can help optimise human productivity and well-being.}} Wisdom gained from a scientific study of motivation and emotion includes that: # '''Motivation''' = behavioural energy, direction, and persistence # '''What we don't know''' about motivation and emotion exceeds what we do know # The '''brain''' is as much about motivation and emotion as it is about cognition and thinking # We underestimate how powerful a motivational force '''biological urges''' can be when not experiencing them # '''Quality''' of motivation (intrinsic vs. extrinsic) is important # To flourish, motivation needs supportive conditions, especially '''supportive relationships''' <!-- # '''Implicit''' (unconscious) '''motives''' predict behaviour better than explicit (conscious) motives --> # We do our best when we have a specific plan of action to pursue a '''difficult, specific, and self-congruent goal''' # People with '''different mindsets''' pursue goals in different ways # The core '''self-efficacy beliefs''' of "I can do it" and "it will work" underlie competent functioning # Exert '''self-control over short-term urges''' to effectively pursue long-term goals # All '''emotions are good''' because they serve a functional purpose # '''Other people''' are the source of most of our emotions # The more sophisticated our '''emotional repertoire''', the more likely we are to have the right emotions in every situation # '''Encouraging growth''' is more productive than trying to cure weakness # Motivation often arises from outside of '''conscious awareness''' # There is nothing so practical as a '''good theory''' ==Readings== * Chapter 16: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) ==Slides== * [https://docs.google.com/presentation/d/1rpaBEkSs5p0TFT3i5JT_o0vG3wZkTJtFAdMO74h8teQ/edit?usp=sharing Interventions] (Google Slides) * [https://docs.google.com/presentation/d/1kucigYPF-72cP0FAbzQZmnOUhNfilMSsbGOQnrnWRaM/edit?usp=sharing Review] (Google Slides) <!-- * [http://www.slideshare.net/jtneill/conclusion-and-review Lecture slides] (Slideshare, 2016) * Handouts **[[Media:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|100px]] * 2014 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] * 2011 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ==Recordings== ;13/11/2013 {{Motivation and emotion/Lectures/Recording3|2013|5b4fdfbe-0073-498f-ad85-53d9c6d06e89}} ;16/11/2011 {{Motivation and emotion/Lectures/Recording|2011|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.m4v|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.mp3}} <!-- * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.mp3 mp3] (audio) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.m4v m4v] (video) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872 echo360] (rich media) --> ==See also== * [[/Images/]] ; Lecture * [[{{#titleparts:{{PAGENAME}}|2}}/Growth psychology|Growth psychology]] (Previous lecture) ; Tutorial * [[Motivation and emotion/Tutorials/Review|Review]] (Tutorial) ==Recording== * [https://au-lti.bbcollab.com/recording/b13748d85c474a87a061ab86fdc19adc Lecture 12] (2025)<!-- * [https://au-lti.bbcollab.com/recording/daa1ba979ba045799efbe73f3226b00f Lecture 12] (2024) * [https://au-lti.bbcollab.com/recording/8ef2ff0521b1457197d7920a93cffc99 Lecture 12] (2023) * [https://au-lti.bbcollab.com/recording/53b16231d9904444bac31b14bc7784f6 Lecture 12 recording] (2022) * [https://au-lti.bbcollab.com/recording/0bd459db9ece46b388a4b794f5751cff Lecture 12 recording] (2021) * [https://au-lti.bbcollab.com/recording/e810c154cfbe41189f8fb4056c871206 Lecture 12 recording] (2020) * [https://echo360.org.au/media/ea2a5707-6be8-43fc-996c-e176ce54bc4f/public Lecture 12 recording] (2018) * [https://echo360.org.au/media/2417e774-f589-4635-9955-9aaf73bcd3b8/public Lecture 12 recording] (2017)--> ==References== {{Hanging indent|1= Cheon, S. H., Reeve, J., & Moon, I. S. (2012). Experimentally based, longitudinally designed, teacher-focused intervention to help physical education teachers be more autonomy supportive toward their students. ''Journal of Sport and Exercise Psychology'', ''34''(3), 365–396. https://doi.org/10.1123/jsep.34.3.365 Izard, C. E., King, K. A., Trentacosta, C. J., Morgan, J. K., Laurenceau, J.-P., Krauthamer-Ewing, S. E., & Finlon, K. J. (2008). Accelerating the development of emotion competence in Head Start children: Effects on adaptive and maladaptive behavior. ''Development and Psychopathology'', ''20'', 369–397. https://doi.org/10.1017/S0954579408000175[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/82060B84CF144DBC4E1AC53CEC26A494/S0954579408000175a.pdf/accelerating_the_development_of_emotion_competence_in_head_start_children_effects_on_adaptive_and_maladaptive_behavior.pdf] Yeager, D. S., Trzesniewski, K. H., & Dweck, C. S. (2013). An implicit theory of personality intervention reduces adolescent aggression in response to victimization and exclusion. ''Child Development'', ''84'', 970–988. https://doi.org/10.1111/cdev.12003 }} <!-- ==External links== --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Interventions and review]] d8i3lz3kigkvxrhu0cawvfvp9n0vytz 2844176 2844175 2026-11-03T11:13:03Z Jtneill 10242 Update for 2026 2844176 wikitext text/x-wiki {{Motivation and emotion/Lectures|Lecture 12: Interventions and review|twelfth and final}} {{Motivation and emotion/Lectures/Complete}} <!-- {{Motivation and emotion/Lectures/In development}} --> <!-- {{Motivation and emotion/Lectures/Complete}} --> [[File:Childcare Development Center-Crestwood High School cheerleaders 120815-F-PG936-400.jpg|right|250px|thumb|'''Figure 1'''. Interventions to change motivation and emotion are commonly used in educational settings.]] ==Overview== This lecture applies and reviews the [[motivation and emotion]] unit by discussing: * interventions using motivation and emotion principles * key points and wisdom gained Take-home messages: * Motivation guides engagement in behaviours that optimise well-being * Emotions provide feedback about progress towards goals * There is nothing so practical as a good theory ==Outline== * Interventions ** Interventions ** Practical problems ** Example interventions * Review ** Unit aims ** Nuggets of truth / Wisdom gained ** Review of key content ==Interventions== The ultimate test of a science is how well its principles can be applied to change desired outcomes. From a psychological point of view, motivation and emotion are dynamic and malleable and, therefore, can also be intentionally changed and strengthened. Well-designed interventions can develop motivational and emotional resources which help promote life outcomes that people care about (e.g., engagement, learning, performance, and well-being) (see Figure 1). The "trick" is not to try changing outcomes directly, but rather to ''alter the conditions'' so they are more conducive to the emergence of the desired outcomes. ==Plan== To systematically intervene, create and follow a plan: # '''Explain''': Diagnose–''Why'' is this happening? What is motivating this person? (Remember: ''All'' behaviour is motivated.) # '''Predict''': What will happen if nothing is done? What could happen if X changed? What if Y or Z changed? Consider options. # '''Intervene''': Apply motivation and emotion principles. Implement a step-by-step plan to alter: #* environment #* relationships #* motivation #* emotion # '''Observe/measure/adjust''': #* Measure key processes and outcomes (before, during, after) #* Track progress regularly #* Adjust strategies based on evidence of what works or not #* Document what was done, what changes, and why Change is difficult. Complex problems often need persistence, feedback, and refinement. If an intervention doesn't work, analyse why, and try a revised approach. ==Experimental manipulations== <div align="center"> {| class="wikitable mw-collapsible" |+ Examples of brief experimental manipulations ! Ch ! Motivation/Emotion ! Experimental manipulation |- | 3 | Trust | Oxytocin hormone via nasal spray |- | 4 | Sexual attraction | Facial metrics (large eyes, small nose, small chin) |- | 5 | Extrinsic motivation | Offering a scholarship for high grades |- | 6 | Autonomy | Self-direction in pursuit of a personal goal |- | 7 | Discrepancy | A difficult, specific, self-concordant goal to strive for |- | 8 | Growth mindset | Story about how Einstein worked hard to become smart |- | 9 | Self-efficacy | A highly competent role model |- | 10 | Self-control depletion | Resisting an attractive temptation for 5 minutes |- | 11 | Positive affect | Receiving a small, unexpected gift |- | 12 | Sadness | Viewing a film about a son at his Dad’s funeral |- | 13 | Embarrassment | Committing a social blunder in front of an audience |- | 14 | Congruence | Unconditional positive regard |- | 15 | Obsession | Keep a secret about the person you are talking to |} </div> ''Note''. Based on Reeve (2024), Table 16.2, p. 350 ==Mini-theories== Use theory to help explain the cause of what is happening and possible forms of intervention. All theories are (potentially) "correct"—the practitioner's job is to match the moment (time, place, context) with a potentially useful mini-theory. Here's a list of some key mini-theories discussed in this unit: * Achievement motivation * Arousal * Attribution * Basic psychological needs * Broaden-and-build * Cognitive dissonance * Cognitive evaluation theory * Drive * Ego depletion * Ego development * Facial feedback * Flow * Fully functioning person * Goal setting * Intrinsic-extrinsic motivation * Learned helplessness * Mindsets * Positive affect * Psychodynamics * Reactance * Self-actualisation * Self-concordance * Self-determination * Self-efficacy * Stress and coping What theories could help predict motivational and/or emotional responses to the following conditions? * Deprivation from food and people * Seeing a threatening (angry) face * Smelling rotting food * Encountering an obstacle to one's goal * Expecting a tangible reward (e.g., money) * Uncontrollable environment * Failure in front of others * Observing a competent role model * Autonomy-supportive teacher ==Intervention strategies== There are two basic strategies for intervening: * '''Overcome''': Fix what isn't working ** Repair weaknesses ** Overcome pathology * '''Promote''' and further develop what is already working ** Amplify strengths ** Improve functioning ==Practical problems== What intervention ideas, based on motivation or emotion theory, can you generate for the following scenarios? * Child not wanting to brush teeth * Sales employee quota * Elite athlete performance * Weight loss * Study motivation These scenarios are explained in more detail below (based on Reeve, 2024, Box 16, p. 348) Guide: * Why? (Explain) * What will happen; what could be? (predict) (Predict) * Implementation (application) of strategies (Intervene) Explain → Predict → Intervene: {{RoundBoxTop|theme=3}} ;Child not wanting to brush teeth A 6-year girl resists brushing her teeth every night before bed. She finds the routine boring and doesn't understand why it's important. When asked to brush her teeth, she plays with the water and brushes for only a few seconds. Her parents, aware of the long-term benefits of good dental hygiene, find themselves in a constant battle to get her to complete the task properly. None of their strategies seem to work. This nightly conflict drains the family's energy and patience. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Sales employee quota A mid-level sales representative at a large company feels increasingly disillusioned with her role. Despite knowing she has the skills to sell, she faces rejection on nearly every call. Her company only measures her success through monthly quotas, providing no support for personal development or job satisfaction. She meets her quota some months, but the constant rejection and lack of recognition make her feel drained. Her confidence has eroded, and she is beginning to question whether this job aligns with her strengths and interests. She fantasises about quitting but worries about financial stability. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ; Elite athlete performance A promising 18-year-old athlete (or musician) is in love with her craft. She spends hours practicing, enjoys the discipline, and consistently performs well. Despite her dedication, her growth in skill level has stagnated over the past year. While her peers are progressing rapidly, she feels stuck. Coaches and teachers praise her current ability but offer no new insights on how to break through to the next level. Frustrated, she worries that no matter how hard she works, she won't reach her goal of becoming an elite performer. Her motivation remains high, but she fears plateauing permanently. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Weight loss A middle-aged man has been advised by his physician to lose 15 kilograms to avoid serious health risks, including a possible heart attack. He acknowledges the logic of the advice but feels disconnected from the urgency of the situation. A long history of poor dietary habits and a sedentary lifestyle has left him with little confidence that he can make meaningful changes. Despite multiple attempts to exercise and eat healthier in the past, nothing has ever stuck. He feels overwhelmed by the prospect of overhauling his lifestyle and is pessimistic that the results would justify the effort. {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} ;Study motivation A university student in his third year is taking an advanced course in a subject he thought he could handle. Despite attending lectures and completing the readings, he performed poorly on the mid-term exam, far below his usual standards. The grade was a shock, leaving him feeling embarrassed and ashamed. He begins to doubt his abilities and contemplates giving up entirely. Each new lecture feels more overwhelming, and his efforts in studying feel futile. As the second half of the semester looms, his motivation plummets, and he wishes he could just drop the course. {{RoundBoxBottom}} ==Nuggets of truth== [[File:Pearl 1.svg|right|50px]] [[File:Treasurchest.svg|right|50px]] After a tsunami of theories and research, let's gather your pearls or wisdom or nuggets of truth about the psychology motivation and emotion: * What are the '''greatest insights''', '''significant learnings''', or '''best ideas''' you've acquired through this unit? * What was the '''take-home message'''(s) from the chapter you created? Contribute your insights to: * [[Motivation and emotion/Nuggets of truth|Nuggets of truth]] (Wikiversity) <!-- Example book chapter take-home messages: * [https://x.com/jtneill/status/1451068316131287041 X] (2021) * [[Motivation and emotion/Book/2011/Pearls_of_wisdom|Wikiversity]] (2011) --> ==Wisdom gained== {{Nutshell| Motivation and emotion fuel human behaviour, shaped by unconscious drives and conscious goals, with social relations playing a key role. While we may underestimate the power of biological urges and emotional dynamics, intrinsic motivation and clear plans unlock our best performance. Self-efficacy and self-control help balance short-term impulses with long-term aspirations. The development of emotional intelligence helps with emotion regulation. Expressing and pursing higher needs guides us towards growth, flourishing, and fulfilling our potential. Understanding the functional complexity of motivation and emotion can help optimise human productivity and well-being.}} Wisdom gained from a scientific study of motivation and emotion includes that: # '''Motivation''' = behavioural energy, direction, and persistence # '''What we don't know''' about motivation and emotion exceeds what we do know # The '''brain''' is as much about motivation and emotion as it is about cognition and thinking # We underestimate how powerful a motivational force '''biological urges''' can be when not experiencing them # '''Quality''' of motivation (intrinsic vs. extrinsic) is important # To flourish, motivation needs supportive conditions, especially '''supportive relationships''' <!-- # '''Implicit''' (unconscious) '''motives''' predict behaviour better than explicit (conscious) motives --> # We do our best when we have a specific plan of action to pursue a '''difficult, specific, and self-congruent goal''' # People with '''different mindsets''' pursue goals in different ways # The core '''self-efficacy beliefs''' of "I can do it" and "it will work" underlie competent functioning # Exert '''self-control over short-term urges''' to effectively pursue long-term goals # All '''emotions are good''' because they serve a functional purpose # '''Other people''' are the source of most of our emotions # The more sophisticated our '''emotional repertoire''', the more likely we are to have the right emotions in every situation # '''Encouraging growth''' is more productive than trying to cure weakness # Motivation often arises from outside of '''conscious awareness''' # There is nothing so practical as a '''good theory''' ==Readings== * Chapter 16: Interventions ([[Motivation and emotion/Readings/Textbooks/Reeve/2024|Reeve, 2024]]) ==Slides== * [https://docs.google.com/presentation/d/1rpaBEkSs5p0TFT3i5JT_o0vG3wZkTJtFAdMO74h8teQ/edit?usp=sharing Interventions] (Google Slides) * [https://docs.google.com/presentation/d/1kucigYPF-72cP0FAbzQZmnOUhNfilMSsbGOQnrnWRaM/edit?usp=sharing Review] (Google Slides) <!-- * [http://www.slideshare.net/jtneill/conclusion-and-review Lecture slides] (Slideshare, 2016) * Handouts **[[Media:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Interventions and review 3slidesperpage.pdf|100px]] * 2014 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 6slidesperpage.pdf|100px]] * 2011 handouts ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|Download 6 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ** [[Media:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 3slidesperpage.pdf|Download 3 slides to a page]]: [[File:Motivation and Emotion - Lecture 12 - Conclusion and review 2011 6slidesperpage.pdf|100px]] ==Recordings== ;13/11/2013 {{Motivation and emotion/Lectures/Recording3|2013|5b4fdfbe-0073-498f-ad85-53d9c6d06e89}} ;16/11/2011 {{Motivation and emotion/Lectures/Recording|2011|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.m4v|http://echo360access.canberra.edu.au/ess/echo/presentation/2f0f632f-3432-4422-a0d3-20db5bbea67f/media.mp3}} <!-- * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.mp3 mp3] (audio) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872/media.m4v m4v] (video) * [http://echo360access.canberra.edu.au/ess/echo/presentation/1939052c-feca-4ab2-8c81-4b28bb445872 echo360] (rich media) --> ==See also== * [[/Images/]] ; Lecture * [[{{#titleparts:{{PAGENAME}}|2}}/Growth psychology|Growth psychology]] (Previous lecture) ; Tutorial * [[Motivation and emotion/Tutorials/Review|Review]] (Tutorial) ==Recording== * [https://au-lti.bbcollab.com/recording/b13748d85c474a87a061ab86fdc19adc Lecture 12] (2025)<!-- * [https://au-lti.bbcollab.com/recording/daa1ba979ba045799efbe73f3226b00f Lecture 12] (2024) * [https://au-lti.bbcollab.com/recording/8ef2ff0521b1457197d7920a93cffc99 Lecture 12] (2023) * [https://au-lti.bbcollab.com/recording/53b16231d9904444bac31b14bc7784f6 Lecture 12 recording] (2022) * [https://au-lti.bbcollab.com/recording/0bd459db9ece46b388a4b794f5751cff Lecture 12 recording] (2021) * [https://au-lti.bbcollab.com/recording/e810c154cfbe41189f8fb4056c871206 Lecture 12 recording] (2020) * [https://echo360.org.au/media/ea2a5707-6be8-43fc-996c-e176ce54bc4f/public Lecture 12 recording] (2018) * [https://echo360.org.au/media/2417e774-f589-4635-9955-9aaf73bcd3b8/public Lecture 12 recording] (2017)--> ==References== {{Hanging indent|1= Cheon, S. H., Reeve, J., & Moon, I. S. (2012). Experimentally based, longitudinally designed, teacher-focused intervention to help physical education teachers be more autonomy supportive toward their students. ''Journal of Sport and Exercise Psychology'', ''34''(3), 365–396. https://doi.org/10.1123/jsep.34.3.365 Izard, C. E., King, K. A., Trentacosta, C. J., Morgan, J. K., Laurenceau, J.-P., Krauthamer-Ewing, S. E., & Finlon, K. J. (2008). Accelerating the development of emotion competence in Head Start children: Effects on adaptive and maladaptive behavior. ''Development and Psychopathology'', ''20'', 369–397. https://doi.org/10.1017/S0954579408000175[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/82060B84CF144DBC4E1AC53CEC26A494/S0954579408000175a.pdf/accelerating_the_development_of_emotion_competence_in_head_start_children_effects_on_adaptive_and_maladaptive_behavior.pdf] Yeager, D. S., Trzesniewski, K. H., & Dweck, C. S. (2013). An implicit theory of personality intervention reduces adolescent aggression in response to victimization and exclusion. ''Child Development'', ''84'', 970–988. https://doi.org/10.1111/cdev.12003 }} <!-- ==External links== --> {{Motivation and emotion/Lectures/Navigation}} [[Category:Motivation and emotion/Lectures/Interventions and review]] 14s17245r9i9vtlwfl3zcg5pv846817 0 104030 2844187 2843314 2026-11-03T11:58:09Z The Citer 3110681 Added categories! 2844187 wikitext text/x-wiki <div class="thumb tleft" style="border: 1px solid #CCCCCC; margin:0.5em;"><table border=0><tr><td width=10><p> </p><td> {| class="wikitable" |- | colspan="5" align="center" style="color: black; background: #ccccff" | '''Orders of Mammals''' |- ! align="center" style="color: black; background: #ccccff" |Infraclass ! align="center" style="color: black; background: #ccccff" |Superorder ! align="center" style="color: black; background: #ccccff" |Order ! align="center" style="color: black; background: #ccccff" |Common Representatives |- ||Eutheria||Afrotheria|| Afrosoricida || golden moles |- ||Eutheria||Afrotheria|| Hyracoidea || hyraxes |- ||Eutheria||Afrotheria|| Macroscelidea || elephant shrews |- ||Eutheria||Afrotheria|| Sirenia || manatees |- ||Eutheria||Afrotheria|| Proboscidea || elephants |- ||Eutheria||Afrotheria|| Tubulidentata || aardvarks |- |||||||| |- ||Eutheria||Euarchontoglires|| Dermoptera || flying lemurs |- ||Eutheria||Euarchontoglires|| Lagomorpha || rabbits |- ||Eutheria||Euarchontoglires|| Primates || chimps |- ||Eutheria||Euarchontoglires|| [[/Rodentia/]] || mice |- ||Eutheria||Euarchontoglires|| Scandentia || tree shrews |- |||||||| |- ||Eutheria||Laurasiatheria|| Artiodactyla || pigs, goats, sheep, cows |- ||Eutheria||Laurasiatheria|| Carnivora || cats and dogs |- ||Eutheria||Laurasiatheria|| Cetacea || whales and dolphins |- ||Eutheria||Laurasiatheria|| Chiroptera || bats |- ||Eutheria||Laurasiatheria|| Erinaceomorpha || hedgehogs |- ||Eutheria||Laurasiatheria|| Perissodactyla || horses |- ||Eutheria||Laurasiatheria|| Pholidota || scaly anteaters |- ||Eutheria||Laurasiatheria|| Soricomorpha || white-toothed shrews |- |||||||| |- ||Eutheria||Xenarthra|| Cingulata || armadillos |- ||Eutheria||Xenarthra|| Pilosa || giant anteaters |- |||||||| |- |||||||| |- ||Marsupial||Ameridelphia|| Didelphimorphia || Virginia opossums |- ||Marsupial||Ameridelphia|| Paucituberculata || shrew opossums |- |||||||| |- ||Marsupial||Australidelphia|| Dasyuromorphia || Tasmanian devils |- ||Marsupial||Australidelphia|| Diprotodontia || kangaroos |- ||Marsupial||Australidelphia|| Microbiotheria || monito del monte |- ||Marsupial||Australidelphia|| Notoryctemorphia || marsupial moles |- ||Marsupial||Australidelphia|| Peramelemorphia || marsupial omnivores |- |}</td></tr></table></div> Mammals are a class of [[tetrapods]] that include [[camels]], chimpanzees, [[Felidae|cats]] and rats and elephants, goats and sheep, horses and cows. <ref> http://species.wikimedia.org/wiki/Mammalia </ref> The class of mammals can be divided into two main infraclasses: eutheria and the [[marsupials]]. ===EUTHERIA = the crown placental mammals=== Eutheria (Greek: "true beasts") is a group of mammals consisting of the first placental mammal and all of her decendants. The earliest known fossil eutherian, Eomaia, was found in Asia, and is dated to the Early Cretaceous period, about 125 million years ago. <ref> http://en.wikipedia.org/wiki/Eutheria </ref> Most placental mammals belong to one of four main superorders: Afrotheria, Euarchontoglires, Laurasiatheria, and Xenarthra. <div class="thumb tright" style="border: 1px solid #CCCCCC; margin:0.5em;"> {|border="0" cellpadding="2" cellspacing="0" |{{clade| style=font-size:90%;line-height:75% | label1=Eutheria <br> <br> | 1={{clade | label1=Boreoeutheria <br> <br> | 1={{clade | 1=Euarchontoglires [[File:Eastern_Cottontail.JPG|60px]] | 2=Laurasiatheria [[File:Asiatische-Goldkatze-Catopuma-temminckii-tier-katze-0001_2.JPG|60px]]}} | label2=Atlantogenata <br> <br> | 2={{clade | 1=Xenarthra [[File:Bicho-preguiça_3.jpg|60px]] | 2=Afrotheria [[File:African_Elephant_(Loxodonta_africana)_male_(17289351322).jpg|55x60px]]}} }} }} |} </div> Euarchontoglires + Laurasiatheria = Boreoeutheria and Xenarthra + Afrotheria = Atlantogenata. Xenarthra diverged from Afrotheria relatively rapidly after Atlantogenata diverged from Boreoeutheria. There was considerable gene flow among the nascent clades prior to their divergence. This made it difficult to resolve the above phylogenic tree until relatively recently. <ref> http://mbe.oxfordjournals.org/content/24/9/2059.short </ref> '''MARSUPIALS''' The seven living orders of marsupials are divided into two clades: Ameridelphia and Australidelphia. <ref> http://www.ncbi.nlm.nih.gov/pubmed/12878458 </ref> '''MONOTREMATA''' <div class="thumb tright" style="border: 1px solid #CCCCCC; margin:0.5em;"> {|border="0" cellpadding="2" cellspacing="0" |{{clade| style=font-size:90%;line-height:75% | label1= mammals <br> <br> | 1={{clade | label1= Theria <br> <br> | 1={{clade | 1= Eutheria [[File:Elaphurus_davidianus_001.jpg|60px]] | 2=Marsupials [[File:Koala_climbing_tree.jpg|60px]]}} | 2={{clade | 1= Monotremata [[File:Duck-billed_platypus_(Ornithorhynchus_anatinus)_Scottsdale.jpg|60px]] }} }} }} |} </div> Order [[monotremata]] = Subclass prototheria Eutheria + Marsupials = Subclass theria ==Phylogeny== {{Clade | style=font-size:90%;line-height:70% |label1='''Mammalia''' |1={{Clade |label1=Yinotheria |1={{clade |1=†Shuotheriidae [[File:Pseudotribos BW (flipped).jpg|60px|''Pseudotribos robustus'']] |label2=Australosphenida |2={{clade |1=†Ausktribosphenidae |2=†Bishopidae |3=†Henosferidae |label4=Monotremata |4={{clade |1=''Ornithorhynchus [[File:Genera mammalium Ornithorhynchus anatinus.jpg|50 px|''Ornithorhynchus anatinus'']] |label2=Tachyglossidae |sublabel2=(Echidna) |2={{clade |1=''Tachyglossus [[File:Tachyglossus aculeatus side on.jpg|60px|''Tachyglossus aculeatus'']] |label2=''Zaglossus |2={{clade |1=''Zaglossus attenboroughi [[File:Zaglossus attenboroughi Flannery.jpg|60px| ''Zaglossus attenboroughi'']] |2=''Zaglossus bruijnii [[File:Long-beakedEchidna.jpg|60px|''Zaglossus bruijnii'']] |3=''Zaglossus bartoni [[File:Zaglossus bartoni - MUSE.JPG|60px|''Zaglossus bartoni'']]}}}} }}}} }} |label2=Theria |2={{Clade |label1=Marsupialia |1={{Clade |1=Paucituberculata [[File:Phylogenetic tree of marsupials derived from retroposon data (Paucituberculata).png|50 px]] |2={{Clade |1=Didelphimorphia [[File:A hand-book to the marsupialia and monotremata (Plate XXXII) (white background).jpg|50 px]] |label2=Australidelphia |2={{Clade |1=Microbiotheria |label2=Eomarsupialia |2={{Clade |label1=Agreodontia |1={{Clade |1=Notoryctemorphia [[File:Phylogenetic tree of marsupials derived from retroposon data (Notoryctemorphia).png|50 px]] |2={{Clade |1=Peramelemorphia [[File:Phylogenetic tree of marsupials derived from retroposon data (Paramelemorphia).png|50 px]] |2=Dasyuromorphia [[File:Phylogenetic tree of marsupials derived from retroposon data (Dasyuromorphia).png|50 px]] }} }} |2=Diprotodontia [[File:A monograph of the Macropodidæ, or family of kangaroos (9398404841) white background.jpg|50 px|Macropodidae]] }} }} }} }} |label2=Placentalia |2={{Clade |label1=Atlantogenata |1={{Clade |label1=Xenarthra |1={{Clade |1=Cingulata [[File:Nine-banded-Armadillo white background.jpg|50 px|''Dasypus novemcinctus'']] |label2=Pilosa |2={{clade |1=Folivora [[File:Bradypus.jpg|50x50px]] |2=Vermilingua [[File:Natural history of the animal kingdom for the use of young people (Plate XV) (Myrmecophaga tridactyla).jpg|50 px|''Myrmecophaga tridactyla'']]}} }} |label2=Afrotheria |2={{Clade |label1=Paenungulata |1={{Clade |1=Hyracoidea [[File:DendrohyraxEminiSmit white background.jpg|50 px]] |label2=Tethytheria |2={{Clade |1=Sirenia [[File:Manatee white background.jpg|50 px|''Trichechus'']] |2=Proboscidea [[File:Indian elephant white background.jpg|50 px|''Elephas maximus'']] }} }} |label2=Afroinsectiphilia |2={{Clade |1=Tubulidentata [[File:Aardvark2 (PSF) colourised.png|50 px]] |label2=Afroinsectivora |2={{Clade |1=Macroscelidea [[File:Rhynchocyon chrysopygus-J Smit white background.jpg|50 px]] |2=Afrosoricida [[File:Potamogale velox illustration.jpg|50 px]] }} }} }} }} |label2=Boreoeutheria |2={{Clade |label1=Laurasiatheria |1={{Clade |1=Eulipotyphla [[File:Mole white background.jpg|50 px|Talpidae]] |label2=Scrotifera |2={{Clade |1=Chiroptera [[File:Vampire bat white background.jpg|50 px|Desmodontinae]] |label2=Ferungulata |2={{Clade |label1=Ferae |1={{clade |1=Pholidota [[File:FMIB 46859 Pangolin a grosse queue white background.jpeg|50 px|Manidae]] |2=Carnivora [[File:Cynailurus guttata - 1818-1842 - Print - Iconographia Zoologica - Special Collections University of Amsterdam - (white background).jpg|50px|''Acinonyx jubatus'']]}} |label2=Euungulata |2={{Clade |1=Perissodactyla [[File:Equus quagga (white background).jpg|50 px|''Equus quagga'']] |2=Artiodactyla [[File:Walia ibex illustration white background.png|50 px|''Capra walie'']] }} }} }} }} |label2=Euarchontoglires |2={{Clade |1={{Clade |label1=Glires |1={{Clade |1=Lagomorpha [[File:Bruno Liljefors - Hare studies 1885 white background.jpg|50 px|''Lepus'']] |2=Rodentia [[File:Ruskea rotta.png|50 px|''Rattus'']] }} }} |label2=Euarchonta |2={{Clade |1=Scandentia [[File:Die Säugthiere in Abbildungen nach der Natur, mit Beschreibungen (Plate 34) (white background).jpg|50 px]] |label2=Primatomorpha |2={{Clade |1=Dermoptera [[File:Cynocephalus volans Brehm1883 (white background).jpg|50 px]] |2=Primates [[File:Die Säugthiere in Abbildungen nach der Natur, mit Beschreibungen (Plate 8) (white background).jpg|50 px|''Cebus olivaceus'']] }} }} }} }} }} }} }} }} ==References== <references/> [[Category:Eukaryote phylogeny]] [[Category:Animals]] [[Category:Zoology]] [[Category:Biology]] [[Category:Taxonomy]] gd286qiv9fkjv0ruaem2exufa33rhg0 0 121359 2843993 2843410 2026-11-02T17:07:03Z Young1lim 21186 /* Basic Features of VHDL */ 2843993 wikitext text/x-wiki <!----------------------------------------------------------------------> == Flip Flop and Latch == * FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]]) * Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]]) * Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]]) * Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]]) <br> == Versions of VHDL == * VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]]) * VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]]) <br> == Basic Features of VHDL == ==== Data ==== * Data Objects ([[Media:Data.Object.1A.20261102.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]]) * Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]]) * Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]]) * Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]]) * Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]]) <br> ==== Signals & Variables ==== * Signals & Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]]) * Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]]) * Concurrent & Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]]) * Inertial & Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]]) * Simulation & Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]]) <br> ==== Structure ==== * Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]]) * Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]]) * Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]]) </br> ==== Entity and Architecture ==== <br> ==== Block Statement ==== <br> ==== Process Statement ==== <br> ==== Operators ==== <br> ==== Assignment Statement ==== <br> ==== Concurrent Statement ==== <br> ==== Sequential Control Statement ==== <br> ==== Function ==== * Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]]) * Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]]) * Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]]) <br> ==== Procedure ==== <br> ==== Package ==== </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:VHDL]] [[Category:FPGA]] c57uakscxxy1y83xikv7soq6pdvlw49 0 138954 2844186 1490699 2026-11-03T11:55:15Z The Citer 3110681 Updated it! 2844186 wikitext text/x-wiki [[File:Cunaxidae.jpg|thumb|300x300px]] The predacious mite superfamily Bdelloidea (Prostigmata:Acari) is represented by two families, viz Cunaxidae and Bdellidae. The family Cunaxidae is the largest of the two families and is represented by 5 subfamilies, 27 genera and 403 spesies. Their raptorial palpi may be 3-,4- or 5-jointed. Subfamilies Bonziinae den Heyer, Coleoscirinae den Heyer and Cunaxinae Oudemans all have 5-jointed palps; subfamily Scirulinae Den Heyer have 4 -jointed and the subfamily Cunaxoidinae Den Heyer 3-jointed palps. They are medium to fast movers. Cunaxidae have a cosmopolitan distribution, occurring in topsoil, litter and on plants. In their embryological development they go through a 6-legged larvae> 8- legged protonymph> deutonymph> tritonymph> 8 legged adults. They feed on fungi, eelworms, other mites and small insects. Size (with extended palpi) up to 1100 microns. Colo(u)r mostly orange to reddish brown. ==Distribution== Cunaxids are found all over the world, except for Antarctica. ==References== J den Heyer 1980. ''A classification system for the family Cunaxidae'' (Actinedida).Publications of the University of the North.Series A 23: 1-12. J den Heyer 1981. Systematics of the family Cunaxidae Thor 1902 (Actinedia:Acarida). Publications of the University of North. Series A24 :1-19 [[Category:Biology]] [[Category:Taxonomy]] [[Category:Zoology]] [[Category:Animals]] jspq6w2kit9oalkfkb3gy28h2sicu0x 0 152431 2844098 2842467 2026-11-03T04:04:59Z Jtneill 10242 /* See also */ * [[Motivation and emotion/Assessment/Chapter/Feedback/2025|2025 feedback]] 2844098 wikitext text/x-wiki {{title|General feedback about<br>book chapters}} {{RoundBoxTop|theme=3}} * General feedback about the [[Motivation and emotion/Book/2025|2025 student-authored book chapters]] * Detailed feedback about each individual chapter is available on the respective talk pages {{RoundBoxBottom}} __TOC__ ==Overall== # The overall quality of chapters was reasonably good, but there was a wide range # The main issues were: ## Unacknowledged and/or poor use of genAI ## Insufficient review and synthesis of the best research about the topic ## Lack of social contributions <!-- # The best chapters will be [https://twitter.com/jtneill/status/1554447127639519232 tweeted in this thread] --> ==Overview== # The best overviews consisted of: ## A scenario/case study in a feature box with a relevant image ## A simple, evocative description of the psychological problem or phenomenon ## Clear and relevant focus questions in a feature box; the focus questions are closely aligned to the top-level headings. ==Theory== # Most chapters used a well selected range of relevant theories # Weaker chapters tended to use more general, historical theories; stronger chapters tended to identify more specific, recent theories # Theories were generally well explained, but theoretical depth and relevance was a key differentiator between chapters # All chapters could be improved by building more strongly on related chapters (i.e., by embedding links from the first mention of key words to related chapters) # More practical examples would have been useful to demonstrate how theories work in practice ==Research== # The best chapters provided comprehensive and critical synthesis of the best research # Most chapters review some relevant research, but could often the research could have been more relevant and/or indepth # Place more emphasis on major reviews such as meta-analyses # Sometimes there was insufficient citation of academic peer-reviewed sources to support claims ==Integration== # The best chapters had excellent integration between theory and research # Many chapters emphasised theory more than research — strive for balance ==Conclusion== # Useful summaries of key points were provided # The best chapters emphasised explicit take-home messages for each focus question ==Style== ===Written expression=== # Overall, the quality of written expression was reasonably good, but varied widely. # The most common problems were: ## grammar and spelling ### poor grammar — there's many ways to get help to develop professional-level written expression (e.g., Studiosity, genAI) ### use 3rd person, rather than 1st person, perspective[https://www.grammarly.com/blog/first-second-and-third-person/] ### use past tense when describing research studies, although implications of findings could be in the present tense ### [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should] should be used (they are part of APA style) ### use of [https://grammar.yourdictionary.com/punctuation/apostrophe-rules.html ownership apostrophes] ### Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling]. ## insufficient citation — scientific communication needs evidence to support claims ## clarity of written expression ### "People" is often a better term than "individuals" ### Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the content/key point to be communicated, with the citation included along the way or, more typically, in parentheses at the end of the sentence. # APA style: ## Citations with three or more authors should use the first author's surname followed by et al. and the year ## Reference formatting areas for improvement: ### capitalisation ### italicisation ### use of hyperlinked dois ===Learning features=== # Embedded interwiki links to Wikipedia articles help to connect reader to further information about key concepts # Embedded interwiki links to related Wikiversity [[Motivation and emotion/Book|book chapters]] were underutilised — network the chapter to related chapters # Figures were reasonably well used, with several students uploading their own images (thankyou!) # Tables were less commonly used, but usually very useful # Feature boxes were generally well used # Quizzes were generally well used # Scenarios/case studies were reasonably well used, but often more examples would have been helpful ==Social contributions== # There were many valuable improvements made to past and current chapters # The amount and quality of contributions varied widely — some students made no contributions whilst some excelled and were awarded bonus marks # Unless there were ''direct links'' to evidence, no marks were provided == See also == * [[Motivation and emotion/Assessment/Chapter/Feedback/2021|2021 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2022|2022 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2023|2023 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2024|2024 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2025|2025 feedback]] * [[Motivation and emotion/Assessment/Chapter|Book chapter - Guidelines]]<!-- * [[Motivation and emotion/Assessment/Multimedia/Feedback|Multimedia presentation feedback]] --> [[Category:Motivation and emotion/Assessment/Chapter]] g06wn9fgs1hbgmfwxyfnodip7c5bjsl 2844103 2844098 2026-11-03T04:28:42Z Jtneill 10242 Update for 2026 2844103 wikitext text/x-wiki {{title|General feedback about<br>book chapters}} {{RoundBoxTop|theme=3}} * General feedback about the [[Motivation and emotion/Book/2026|2026 student-authored book chapters]] * Detailed feedback about each individual chapter is available on the respective talk pages {{RoundBoxBottom}} __TOC__ ==Overall== # The overall quality of chapters was reasonably good, but there was a notably wide range # The main issues were: ## Unacknowledged and/or poor overuse use of genAI ## Insufficient review and synthesis of the best research about the topic ## Minimal social contribution ==Overview== # The best overviews consisted of: ## A scenario/case study which appealed to a broad audience in a feature box with a relevant image ## A straighforward, compelling description of the psychological problem or phenomenon ## Clear and relevant focus questions that closely aligned with the top-level headings. ==Theory== # Most chapters used a well-selected range of relevant theories # Theories were generally well explained, but stronger chapters exhibit more incisive theoretical depth based on reading and understanding of primary sources # Almost chapters could be improved by more strongly embedding links to related chapters # The strongest chapters illustrated theoretical princples through practical examples ==Research== # Depth of critical review of research was the single most distinguishing characteristic of the strongest chapters # Most chapters reviewed some relevant research, but often the review of research could have been more relevant and/or indepth # More emphasis should have been placed on major reviews such as meta-analyses # To improve critical review of research, for key studies, report details about the sample, country, method, and effect sizes # Often there was insufficient citation of academic peer-reviewed sources to support claims ==Integration== # The best chapters had excellent integration between theory and research # Weakers chapters tended to rely more heavily on theory than research, or covered research as a stand-alone topic rather than integrated review that synthesised theory and research ==Conclusion== # Often the weaker conclusions read like generic genAI summaries # The best chapters addressed the focus questions and provided explicit take-home messages ==Style== ===Written expression=== # The overall quality of written expression was reasonably good, but varied widely # The most common problems were: ## grammar and spelling ### poor grammar—there's many ways to get help to develop professional-level written expression (e.g., Studiosity, genAI) ### use 3rd person, rather than 1st person, perspective[https://www.grammarly.com/blog/first-second-and-third-person/] ### use past tense when describing research studies, although implications of findings could be in the present tense ### use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should] (they are part of APA style) ### use [https://grammar.yourdictionary.com/punctuation/apostrophe-rules.html ownership (possessive) apostrophes] ### Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling]. ## insufficient citation—psychological science community requires claims to be supported by cited evidence and sources # The most common APA style reference formatting areas for improvement were: ### capitalisation ### italicisation ===Learning features=== # The use of learning features varied widely between chapters # The best chapters were interconnected with other Wikimedia resources by using: ## embedded interwiki links to Wikipedia articles help to connect reader to further information about key concepts ## embedded interwiki links to related Wikiversity [[Motivation and emotion/Book|book chapters]] # Figures were reasonably well used, with many students uploading and contributing their own images (thankyou!) # Tables were less commonly used, but were strongest when thoughtfully constructed and weaker when constructed by genAI and lacking in acknowledgement of sources # Additional scenarios/case studies in feature boxes were underused to provide illustrative examples # The best chapters used distributed quiz questions about take-home messages rather than being presented at the end ==Social contributions== # The amount and quality of contributions varied widely: The use of social contributions distinguished sharply between genuinely and casually engaged authors—some students made no contributions whilst some excelled and were awarded bonus marks # Many valuable improvements were made to past and current chapters # Unless there were ''direct links'' to evidence, no marks were provided == See also == * [[Motivation and emotion/Assessment/Chapter/Feedback/2021|2021 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2022|2022 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2023|2023 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2024|2024 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2025|2025 feedback]] * [[Motivation and emotion/Assessment/Chapter|Book chapter - Guidelines]]<!-- * [[Motivation and emotion/Assessment/Multimedia/Feedback|Multimedia presentation feedback]] --> [[Category:Motivation and emotion/Assessment/Chapter]] gsljpwbpvmint5yjujxttg4whw7ky5f 2844104 2844103 2026-11-03T04:39:40Z Jtneill 10242 Update 2844104 wikitext text/x-wiki {{title|General feedback about<br>book chapters}} {{RoundBoxTop|theme=3}} * General feedback about the [[Motivation and emotion/Book/2026|2026 student-authored book chapters]] * Detailed feedback about each individual chapter is available on the respective talk pages {{RoundBoxBottom}} __TOC__ ==Overall== # The overall quality of chapters was reasonably good, but there was a notably wide range in quality from poor to outstanding # The most common issues were: ## Not directly addressing the topic ## Making unacknowledged and/or poor overuse use of genAI, rather than prioritising human writing based on reading of original sources ## Insufficient review and synthesis of the best research about the topic (weaker chapters were commonly theory-heavy and research-light) ## No or minimal engagement in making social contributions ==Overview== # The best overviews consisted of: ## A scenario/case study which appealed to a broad audience in a feature box with a relevant image ## A straighforward, compelling description of the psychological problem or phenomenon ## Clear and relevant focus questions that closely aligned with the top-level headings. ==Theory== # Most chapters used a well-selected range of relevant theories # Theories were generally well explained, but stronger chapters exhibit more incisive theoretical depth based on reading and understanding of primary sources # Almost chapters could be improved by more strongly embedding links to related chapters # The strongest chapters illustrated theoretical princples through practical examples ==Research== # Depth of critical review of research was the single most distinguishing characteristic of the strongest chapters # Most chapters reviewed some relevant research, but often the review of research could have been more relevant and/or indepth # More emphasis should have been placed on major reviews such as meta-analyses # To improve critical review of research, for key studies, report details about the sample, country, method, and effect sizes # Often there was insufficient citation of academic peer-reviewed sources to support claims ==Integration== # The best chapters had excellent integration between theory and research # Weakers chapters tended to rely more heavily on theory than research, or covered research as a stand-alone topic rather than integrated review that synthesised theory and research ==Conclusion== # Often the weaker conclusions read like generic genAI summaries # The best chapters addressed the focus questions and provided explicit take-home messages ==Style== ===Written expression=== # The overall quality of written expression was reasonably good, but varied widely # The most common problems were: ## grammar and spelling ### poor grammar—there's many ways to get help to develop professional-level written expression (e.g., Studiosity, genAI) ### use 3rd person, rather than 1st person, perspective[https://www.grammarly.com/blog/first-second-and-third-person/] ### use past tense when describing research studies, although implications of findings could be in the present tense ### use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should] (they are part of APA style) ### use [https://grammar.yourdictionary.com/punctuation/apostrophe-rules.html ownership (possessive) apostrophes] ### Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling]. ## insufficient citation—psychological science community requires claims to be supported by cited evidence and sources # The most common APA style reference formatting areas for improvement were: ### capitalisation ### italicisation ===Learning features=== # The use of learning features varied widely between chapters # The best chapters were interconnected with other Wikimedia resources by using: ## embedded interwiki links to Wikipedia articles help to connect reader to further information about key concepts ## embedded interwiki links to related Wikiversity [[Motivation and emotion/Book|book chapters]] # Figures were reasonably well used, with many students uploading and contributing their own images (thankyou!) # Tables were less commonly used, but were strongest when thoughtfully constructed and weaker when constructed by genAI and lacking in acknowledgement of sources # Additional scenarios/case studies in feature boxes were underused to provide illustrative examples # The best chapters used distributed quiz questions about take-home messages rather than being presented at the end ==Social contributions== # The amount and quality of contributions varied widely: The use of social contributions distinguished sharply between genuinely and casually engaged authors—some students made no contributions whilst some excelled and were awarded bonus marks # Many valuable improvements were made to past and current chapters # Unless there were ''direct links'' to evidence, no marks were provided == See also == * [[Motivation and emotion/Assessment/Chapter/Feedback/2021|2021 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2022|2022 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2023|2023 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2024|2024 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2025|2025 feedback]] * [[Motivation and emotion/Assessment/Chapter|Book chapter - Guidelines]]<!-- * [[Motivation and emotion/Assessment/Multimedia/Feedback|Multimedia presentation feedback]] --> [[Category:Motivation and emotion/Assessment/Chapter]] lzls4w22ffak7pjvnvnzsxxgdqq1jjg 0 170037 2843981 2191019 2026-11-02T14:21:56Z ~2026-58066-80 3113702 link fix 2843981 wikitext text/x-wiki '''IELTS Speaking Topics''' To prevent memorized responses and to make the IELTS speaking component a good test of an IELTS candidate's English level, the makers of this exam have a variety of IELTS questions that they can use for all three parts of the speaking component. In addition, these topics are normally changed a few times per year with new questions added. '''Part 1 IELTS Speaking Topics''' Part 1 IELTS speaking topics are very general questions which normally ask about your personal experiences. The first group of questions always ask about either your work/school experience or your home. Examples of a IELTS speaking questions in part 1 about your home could be: Tell me about your home? What is the favorite room in your home? While still in this part, the examiner will ask you additional questions related to one or two more general topics, including television, hobbies, and birthdays. On any given day, an IELTS examiner has a choice of around 15 different part 1 topics to choose to ask from. They will never ask an individual candidate more than three of them. Here is list of common [http://ieltsspeaking.ca/ielts-speaking-topics/ part 1 IELTS speaking topics and questions]. '''Part 2 IELTS Speaking Topics ''' Part 2 IELTS speaking topics involve candidates to present an answer for one to two minutes. The examiner will allow one minute for the candidate to prepare the answer with a pencil and a piece of paper. The examiner normally expects the IELTS candidate to spend the full two minutes to present a response to this task. A typical Part 2 IELTS speaking topic may look like this: ''Describe a book that you enjoyed reading:'' - ''What kind of book was it?'' - ''What was the book about?'' - ''Why did you enjoy it?'' ''Would you recommend this book to anyone else?'' Just like Part 1, the examiner has a choice of a wide variety of Part 2 IELTS speaking topics to choose from. These topics are changed a few times each year. This link provides a list of common practice [http://ieltsspeaking.ca|IELTS speaking topics for Part 2]. '''Part 3 IELTS Speaking Questions''' Part 3 of the IELTS speaking exam involves discussion related questions that related to the same topic used in Part 2. These questions are much more challenging than Part 1 of the exam as they demand candidates to speak more specifically about more complex ideas outside of their own personal world. If a candidate was asked the 'book' topic in Part 2, they may asked IELTS Part 3 questions, such as: - ''What is better to use an ebook or a paperback book?'' - ''What will libraries be like in the future?'' - ''Why is reading such an important skill'' IELTS Part 3 questions normally ask candidates to predict about the future, make comparisons and evaluations, explain reasons, and talk about the past. 5myxaumodew0riafvo8hgxavuwe6pmj 0 171301 2844086 2843351 2026-11-03T03:08:45Z James500 297601 /* Animation */ Add 2844086 wikitext text/x-wiki {{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}} {{Bibliography}} {{research}} If this resource is ever completed, it will be a universal bibliography.<ref>See [[w:Bibliography]].</ref> Until then, it will be an approximation of a universal bibliography. This bibliography is arranged as an index of topics. ==Index== *[[Universal Bibliography/Bibliography|Bibliography]] *[[Universal Bibliography/Libraries|Libraries]] *[[Universal Bibliography/Literature|Literature]] *[[Universal Bibliography/Languages|Languages]] *[[Universal Bibliography/SF|SF]] *[[Universal Bibliography/Music|Music]] *[[Universal Bibliography/Cinema|Cinema]] *[[Universal Bibliography/Television|Television]] *[[Universal Bibliography/Culture|Culture]] *[[Universal Bibliography/Publishers and imprints|Publishers and imprints]] *[[Universal Bibliography/Printing|Printing]] *[[Universal Bibliography/Printers|Printers]] *[[Universal Bibliography/Microform|Microform]] *[[Universal Bibliography/Periodicals|Periodicals]] *[[Universal Bibliography/Reference|Reference]] *[[Universal Bibliography/Gazetteers|Gazetteers]] *[[Universal Bibliography/Humanities|Humanities]] *[[Universal Bibliography/Law|Law]] *[[Universal Bibliography/History|History]] *[[Universal Bibliography/Archaeology|Archaeology]] *[[Universal Bibliography/Geography|Geography]] *[[Universal Bibliography/Continents|Continents]] *[[Universal Bibliography/Countries|Countries]] *[[Universal Bibliography/Architecture|Architecture]] *[[Universal Bibliography/Mathematics|Mathematics]] *[[Universal Bibliography/Computers|Computers]] *[[Universal Bibliography/Kites|Kites]] *[[Universal Bibliography/Nostalgia|Nostalgia]] *[[Universal Bibliography/Children's non-fiction|Children's non-fiction]] ===About=== *[[Universal Bibliography/About|About]] ==Online libraries== Swedish: *[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank) *[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg) French *[[w:Persée (web portal)|Persée]] (persee.fr) Japanese *[[w:Aozora Bunko|Aozora Bunko]] (aozora.gr.jp) ==Biographical dictionaries etc== See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]] *Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Arthur, "Biographical Dictionaries in the Digital Era". Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&pg=PA83#v=onepage&q&f=false Page 83] et seq. Bibliographies, indexes, etc: *Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ] *Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ] *Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ] *Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ] *Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ] *Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ] *Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ] Particular works: *Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]] *Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&btnG=Search+Books&bksoutput=html_text&tbm=bks&tbo=1 editions:uzt3-qMuFcMC]) *A & C Black's Who's Who *Who Was Who *The Academic Who's Who. A & C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&pg=PA208&output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC *Dictionary of Edwardian Biography (Pike's New Century Series) Works that comprise largely of biographies: *The Penguin Companion to Literature Theatres *A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers & Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15] ==Dictionaries of dates== [https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org] *Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ] *Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&tbm=bks&q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC] *Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880 *[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ] *Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440] *Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930. *[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]] *Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates. *Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will] *Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA93&output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&pg=PA351&output=html_text] *Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May) *Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&id=k3dJAAAAYAAJ&output=html] *Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880] Australia *John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog] *John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford & Co. Brisbane. 1895. America *Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921. ==Commodity dictionaries== *Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&pg=PR22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&pg=RA15-PA7#v=onepage&q&f=false] *Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, "Commodity Dictionary", Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&pg=PA182#v=onepage&q&f=false p 182] & 183 *Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 & 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ] *Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC] *Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&pg=PA78#v=onepage&q&f=false] ==Encyclopedias== See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]] *[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]] *Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990. *The Dorling Kindersley Illustrated Family Encyclopedia ==Almanacs== See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]]. *Year Book and Almanac of Newfoundland. **For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft] *Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C] ==Censuses== *Census of New Zealand and Labrador **1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog] **1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng] **1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng] ==Pilot guides== *[[w:United States Coast Pilot|United States Coast Pilot]] *American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&pg=PR1#v=onepage&q&f=false] *Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ] *Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ] *Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&pg=PR1#v=onepage&q&f=false] *Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&pg=PP7#v=onepage&q&f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&pg=PP5#v=onepage&q&f=false] ==Books of facts== *The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC *Chambers Book of Facts *Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008 *Handy Book of Facts: Things Everyone Should Know. C.S. Hammond & Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ] ==Series of books== See [[w:Category:Series of books]] and [[w:Category:Monographic series]] *George M Sinkankas, "Series" in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&pg=PA250#v=onepage&q&f=false 250] to 273. *"Publishing in Series, 1896-1916" in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&pg=PA539#v=onepage&q&f=false Page 539] et seq. *Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&pg=PP1#v=onepage&q&f=false vol 2]. *Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ] *Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Khan. "Monographs in series". The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&pg=PA208#v=onepage&q&f=false 207] to 209. *Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&pg=PA6#v=onepage&q&f=false 6] and 7. *Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&pg=PA11#v=onepage&q&f=false] **Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&pg=PA1190#v=onepage&q&f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&pg=PA53#v=onepage&q&f=false] ***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ] **Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ] **Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ] **Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ] *Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ] **2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ] **3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA63#v=onepage&q&f=false] *Ocran, Emmanuel Benjamin. Scientific & Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&pg=PA949#v=onepage&q&f=false] *Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ] *Young People's Literature in Series *Catalog of Reprints in Series. (sometimes called "Catalogue of Reprints in Series"). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA73#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ] *Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&pg=PA656#v=onepage&q&f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&pg=PA367#v=onepage&q&f=false] ==Series of classics== *Penguin Classics (Penguin Modern Classics, Penguin English Library) *Oxford World Classics *Everyman's Library *Wordsworth Classics *Macmillan Collectors Library *Bantam Classics *Minster Classics *The Literary Heritage Collection (Heron Books, London. William Collins Sons & Co, Glasgow) *Chandos Classics *Temple Classics *Longmans Heritage of Literature Series Russian *Greatest Masterpieces of Russian Literature (Heron Books, London) SF *Corgi SF Collectors Library Children's and shorter classics etc *Shorter Classics. Ginn and Company. *Ladybird Children's Classics. *Mini Classics. Parragon Books. *Bonny Books. Peter Haddock Ltd. *A series published by Dean & Son Ltd ==Non-fiction general series== *[[w:Oxford Companions|Oxford Companions]] *[[w:Cambridge Companions|Cambridge Companions]] *Princeton Companions *Blackwell Companions. Wiley Blackwell Companions *Routledge Companions. Routledge Research Companions *Ashgate Companions. Ashgate Research Companions *Brill's Companions *Facts on File Companions *Guides to Information Sources. Bowker-Saur *Butterworths Guides to Information Sources. *Columbia Guides *Blackwell Guides *Edinburgh Critical Guides *Collins Reference Dictionaries *New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]]) *Collins Gem (see [[w:List of Collins GEM books]]) *Concise Encyclopedias. Collins. *Time Life Books (see [[w:Time Life#Book series]]) *[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. *[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton. *Made Simple Books. W H Allen. *Palgrave Master Series *Harrap's Mini Series *Shire Albums. Shire Publications. *Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd. *The Wonderful World Books. Macdonald and Company *Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners. *Hamlyn Pocket Guides *Oxford Monograph Series *Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]] *Helpmate Handbooks. Willow Books University *University Paperbacks. Meuthen & Co *World Student Series. Addison Wesley *Unibooks. Hodder and Stoughton *International Student Editions. Van Nostrand Reinhold *Hutchinson University Library Imprints *Pelican Books Pictorials *Salmon Cameracolour series *Pitkin Pictorials United Kingdom *Aspects of Britain. HMSO. Places *The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]] *G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]] Art *Movements in World Art. Meuthen. *Movements in Modern Art. Meuthen. *How to Draw and Paint. New Burlington. Film *BFI Companions Popular science *Contemporary Science Paperbacks. Oliver and Boyd. *Pan Piper Science Series Science and mathematics *Simon and Schuster Tech Outlines *Schaum's Outline Series Military *Illustrated Military Guides. Illustrated Guides. "An Illustrated Guide to ...". Salamander Books. *Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press. *Osprey Men-at-Arms *Jane's Pocket Books Communication *The Library of Communication Techniques. Focal Press. *John Fiske (ed). Studies in Culture and Communication. Routledge. *The Media. Wayland. Cookery *ABC series. Peter Pauper Press. Gardening *Pan Piper Small Gardens Series. Mythology *Series on mythology published by Southwater (imprint of Anness) ==History and Geography== See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]]. *Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&pg=PP1#v=onepage&q&f=false] *Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&pg=PP1#v=onepage&q&f=false] General series *Cambridge Studies in Historical Geography Atlases *The Times Atlas of World History *Philip's Atlas of World History History of geography: *Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ] ==Chronology== See also [[Universal Bibliography/History#Millennia, centuries and decades]] General *Chronology of World History. **Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994. **Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994. **Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994. **Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon & Schuster. Reissued 1994. Centuries *Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ] *Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ] *Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC] *20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ] *Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ] *Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&pg=PP7#v=onepage&q&f=false] Decades *Series: **Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ] Years *Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&pg=PA113#v=onepage&q&f=false Catalogue]. *The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&pg=PP7#v=onepage&q&f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&pg=PP7#v=onepage&q&f=false 1882 to 1883]. *James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&pg=PP7#v=onepage&q&f=false] *[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&pg=PR1#v=onepage&q&f=false 1821]. *Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&pg=PR3#v=onepage&q&f=false] Ancient *Bickerman. Chronology of the Ancient World. 1968. *Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993. ==Anniversaries== *Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ] *On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&pg=PA1#v=onepage&q&f=false] ==Egyptology== *Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&pg=PR3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&pg=PR3#v=onepage&q&f=false] ==Battlefields== *[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series] *"The Times Guide to Battlefields of Britain". Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart. ==Armed forces== Periodicals: *[[w:NATO Review|NATO Review]] Military *The Journal of Military History *Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC] *The Monthly Army List [Google editions:I0t2L4ElznEC] *The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC] *Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC] *The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&q=editions:dEauCcI7kssC&biw=534&bih=736&dpr=1.5#sbfbu=1] *The Royal Tank Corps Journal *The Tank [https://www.google.com/search?sa=N&cs=0&tbm=bks&q=editions:Dv-RbpoM7acC&biw=534&bih=736&dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment *The Cavalry Journal [https://www.google.com/search?sa=N&cs=0&tbm=bks&q=editions:cVQlfkRl6KUC&biw=534&bih=688&dpr=1.5#sbfbu=1] *The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&q=editions:liFy4uc0ggYC&biw=534&bih=736&dpr=1.5] *Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC] *The Royal Engineers Journal [https://www.google.com/search?tbm=bks&q=editions:8XobinXLbD0C&biw=534&bih=736&dpr=1.5] *Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C] *Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC] United States *Military Review *The Coast Artillery Journal [Google editions:nMCogSJ_rlkC] *Infantry Journal [Google editions:ULqoLmbUR5cC] *The Reserve Officer [Google editions:JQDRDrnD1QQC] Naval *[[w:Navy News|Navy News]] ==Armour== Armoured warfare; tank warfare *Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ] *Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ] *Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ] *Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ] *Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ] *Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&pg=PP1#v=onepage&q&f=false] *Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ] Atlas *Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ] Armored forces *Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ] ==Mesoamerica== *James. Aztecs & Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: "Books Received" (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and "Overviews for the general reader" (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252. *Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ] ==Accounting== See [[s:Category:Accounting]] Periodicals *[[s:The Accountant|The Accountant]] (1874 onwards) *Accountant's Magazine (1897 onwards) Aberdeen ==Arts== *Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997. ==Biography== *Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&pg=PP1#v=onepage&q&f=false] *Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Periodicals *Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ] *Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ] Yearbooks *Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC] *Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ] ==Information technology== *Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&pg=PR1#v=onepage&q&f=false] ==Economics== General series: *Dryden Press Series in Economics *Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational *Nuffield Economics & Business. Nuffield Foundation. Longman. Other: *Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984. *Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991. *Anderton, Alain. Economics. Causeway Press. 1991. *Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983. *Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991. *Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986. *Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989. *Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985.  *Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981. *National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990. *Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992. ==Games== Chess *Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987. *Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980. *Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977. *Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. General series *Batsford Chess Books **Discovering Chess Series. B T Batsford. Periodicals See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]] *British Chess Magazine Othello *Goro Hasegawa with Maxine Brady. How to Win at Othello. (A Harvest/HBJ book). Jove Publications. 1977. ISBN 0-15-642215-8. [https://books.google.co.uk/books?id=mTdhlwVeyCcC] Wargames *Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward. **Laurie Taylor. "Attila the Hun invades Tyne Tees". TV Times. 1978. pp 28 & 29. **Terry Wise. "Battleground". Battle for Wargamers. June 1978. pp 261 & 262. *[[w:Game of War|Game of War]]. Channel 4. 1997. [Television]. ==Toys== Periodicals *Games & Toys: The Leading Trade Journal for Home & Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C *Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. "Harcourt To Buy Journals From Haire Publishing Co" [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29 *[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. "Playthings bought by Geyer-McAllister" [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing Capsule toys Gachapon *Tim Hornyak. [https://www.japantimes.co.jp/life/2017/08/19/lifestyle/gachapon-tracing-evolution-japans-colorful-toy-capsules/ Gachapon: Tracing the evolution of Japan’s colorful toy capsules]. The Japan Times. 19 August 2017. ==Cricket== See [[w:Bibliography of cricket]] *Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ]. **Peter Arnold. The Illustrated Encyclopedia of World Cricket. *Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ] Scores and biographies *Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&pg=PR3#v=onepage&q&f=false] **See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]] Periodicals *[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&pg=PP7#v=onepage&q&f=false]. Australia *Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463] *The Oxford Companion to Australian Cricket India *The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies] Annuals *[[w:Indian Cricket (annual)|Indian Cricket]]. [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966]. ===Cricketers=== Cricketers, including biographical dictionaries and collections of biographies *[[w:ESPNcricinfo|ESPNcricinfo]] *[[w:CricketArchive|CricketArchive]] *John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ] *World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ] *Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ] *Sproat. Debrett's Cricketers' Who's Who. 1980. *S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934. *Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ] *Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ] *Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC] *Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ] Gloucestershire *Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html] *Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C]. *Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795] Australia *The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ] *Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ] *C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894. *Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus & Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ]. ==Geology== *Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History. ==Mineralogy== *Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&pg=PA473#v=onepage&q&f=false 473] to 476. *Battey, Maurice Hugh. Mineralogy for students. Oliver & Boyd. 1972. 2nd Ed. Longman. 1981. ==Paper== See [[s:Category:Paper]] *Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf] *West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog] ==Books== *British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ] *Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ] *Book News. 1882 to 1918. (John Wanamaker). Called "Book News Monthly" from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&pg=PP7#v=onepage&q&f=false] *Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ] *U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ] *Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ] *Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ] *Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C] *Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ] Rare *Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ] Printed *Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&pg=PP1#v=onepage&q&f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&pg=PP1#v=onepage&q&f=false vol 31] ==Paperback and Paperbound== *Swados, "Paper Books: What do they Promise?" (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114 *Wagman, "The Paperbound Book Business" (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&pg=RA4-PA9#v=onepage&q&f=false 9] (No 5, November) ==Science== *Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998. ==Entertainment== *The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/] ==Fiction== ===Characters=== *Bowker's Guide to Characters in Fiction [https://books.google.co.uk/books?id=rVjlAAAAMAAJ] *William Freeman. Dictionary of Fictional Characters. [https://books.google.co.uk/books?id=sRYuAAAAMAAJ] *Martin Seymour-Smith. Dictionary of Fictional Characters. The Writer, Inc. [https://books.google.co.uk/books?id=MoQYAAAAIAAJ] *Cyclopedia of Literary Characters. Salem Press. Rev Ed: 1998. [https://books.google.co.uk/books?id=IJMYAAAAIAAJ] *Larousse Dictionary of Literary Characters. [https://books.google.co.uk/books?id=hq4UAQAAIAAJ] *Chambers Dictionary of Literary Characters. [https://books.google.com/books?id=KWNMPwAACAAJ] *Michael D Sollars. Dictionary of Literary Characters. [https://books.google.co.uk/books?id=IJddAQAACAAJ] ==Comedy and tragedy== *E H Mikhail. Comedy and Tragedy: A Bibliography of Critical Studies. 1972. [https://books.google.co.uk/books?id=5bgEAAAAMAAJ] ===Comedy=== *James E Evans. Comedy: An Annotated Bibliography of Theory and Criticism. 1987. [https://books.google.co.uk/books?id=s9ZgthAs1t8C&pg=PP1#v=onepage&q&f=false] *Meghan Duffy (ed). Daniel Gerould (senior ed). Comedy: A Bibliography of Critical Studies in English on the Theory and Practice of Comedy in Drama, Theatre, and Performance. New York. 2006. [https://books.google.co.uk/books?id=fI0jAQAAIAAJ] *David Humphrey. The Time of Laughter: Comedy and the Media Cultures of Japan. University of Michigan Press. 2023. [https://books.google.co.uk/books?id=CkTLEAAAQBAJ&pg=PP1#v=onepage&q&f=false] ==Radio and television== *William E McCavitt. Radio and Television: A Selected Annotated Bibliography. Scarecrow Press. 1978. [https://books.google.com/books?id=DAQVAQAAIAAJ] *Gertrude G Broderick. Radio and Television Bibliography. Bulletin 1948 No 17. [https://books.google.co.uk/books?id=yIPP1dsmNtAC&pg=PP5#v=onepage&q&f=false] *Patricia Beall Hamill. Radio and Television: A Selected Bibliography. Bulletin 1960 No 25. [https://books.google.co.uk/books?id=AJhGAQAAMAAJ&pg=PP3#v=onepage&q&f=false] ==Film and television== *Jean Mitry. Bibliographie internationale du cinéma et de la télévision. 1966 onwards. [https://books.google.co.uk/books?id=SC3gAAAAMAAJ] Commentary: A Reference Guide for English Studies, [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA439#v=onepage&q&f=false p 439]. ==Broadcasting== *Iwao Nakajima. The Broadcasting Industry in Japan: Its Historical, Legal and Economic Aspects. (International Studies of Broadcasting). NHK Radio and TV Culture Research Institute. 1971. [https://books.google.co.uk/books?id=uZ4vAQAAIAAJ] *50 Years of Japanese Broadcasting. NHK. 1977. [https://books.google.com/books?id=FoSKqY7tG-AC] *The History of Broadcasting in Japan. Nippon Hoso Kyokai. 1967. [https://books.google.com/books?id=I1IHAQAAIAAJ] *Hitoshi Mitomo and Mikio Kimura. Broadcasting in Japan: Challenges and Opportunities. [https://books.google.co.uk/books?id=7hCcEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Broadcasting in Japan: Case-studies on Broadcasting Systems. 1978. 2011. [https://books.google.co.uk/books?id=Ez5ZBwAAQBAJ&pg=PR4#v=onepage&q&f=false] *"World's First TV Stereo Broadcasts in Japan". Japan Report. (vol 25, no 3). 1 February 1979. p 4. *Broadcasting in Japan: A Historical Review [https://books.google.com/books?id=uDYlAAAAMAAJ] Annuals *[[w:ja:日本民間放送年鑑|日本民間放送年鑑]] (Japan Commercial Broadcasting Yearbook). コーケン出版. [https://books.google.co.uk/books?id=vh22AAAAIAAJ 1997]. ==Animation== *John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005. *Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&pg=PA1958#v=onepage&q&f=false]. *[https://www.bloomberg.com/news/articles/2026-10-07/anime-s-popularity-is-rising-among-teen-fans Anime is winning over a new generation of fans]. Bloomberg. 9 October 2026. [https://www.japantimes.co.jp/culture/2026/10/09/entertainment-news/anime-survey-watching/ The Japan Times]. ==Colours== *Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/] ==Bilateral== Britain and Japan *Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ] ==Prehistoric life== Prehistoric animals *[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.] *Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&pg=PA269#v=onepage&q&f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&pg=PA5#v=onepage&q&f=false]. **Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977. Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ] **Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme. Dinosaurs *Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.] ==Regions and other places== *JD Pearson. Oriental and Asian Bibliography: An Introduction with Some Reference to Africa. Archon Books. [https://books.google.com/books?id=P18ZAAAAMAAJ] *Theodore Besterman. A World Bibliography of Oriental Bibliographies. Revised by JD Pearson. Rowman and Littlefield. [https://books.google.co.uk/books?id=dvnfAAAAMAAJ] ==Culture and society== *Victoria Lyon Bestor, Theodore C Bestor and Akiko Yamagata (eds). Routledge Handbook of Japanese Culture and Society. 2011. [https://books.google.co.uk/books?id=0cBYffHp5L4C&pg=PP1#v=onepage&q&f=false] *Josef Kreiner and Hans-Dieter Ölschleger (eds). Japanese Culture and Society: Models of Interpretation. 1996. [https://books.google.co.uk/books?id=6skwAQAAIAAJ] *Dolores P Martinez. Modern Japanese Culture and Society. 2007. [https://books.google.co.uk/books?id=l6kMAQAAMAAJ vol 1] *C Andrew Gerstle. 18th Century Japan: Culture and Society. 1989. [https://books.google.co.uk/books?id=ViglDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Periodicals *Review of Japanese Culture and Society [https://books.google.co.uk/books?id=W_o-AQAAIAAJ] ==Clothes== *John Marshall. Make Your Own Japanese Clothes: Patterns and Ideas for Modern Wear. Kodansha International. 1988. [https://books.google.co.uk/books?id=ljOuuy0OO0UC&pg=PP1#v=onepage&q&f=false] ==Fashion== *Toby Slade. Japanese Fashion: A Cultural History. 2009. [https://books.google.co.uk/books?id=AInI19E48QIC&pg=PP3#v=onepage&q&f=false] *Bonnie English. Japanese Fashion Designers: The Work and Influence of Issey Miyake, Yohji Yamamotom, and Rei Kawakubo. 2011. [https://books.google.co.uk/books?id=6UMfAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sadao Hibi. Japanese Detail Fashion. Chronicle Books. 1989. [https://books.google.com/books?id=G-kIAQAAMAAJ] **Nihon No Dento Iro To Katachi. Graphic-sha Publishing. 1987. Kimono *Sheila Cliffe. The Social Life of Kimono: Japanese Fashion Past and Present. 2017. [https://books.google.co.uk/books?id=7G6xDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Anna Jackson (ed). Kimono: The Art and Evolution of Japanese Fashion. V&A Publishing. 2010. [https://books.google.co.uk/books?id=8gtMAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/kimono-the-art-and-evolution-of-japanese-fashion-the-khalili-collections/oclc/917375803]. ==See also== *[[Bibliography]] ==Notes== {{Reflist}} {{subpagesif}} [[Category:Bibliographies]] [[Category:Research]] fjrytji8y90tf6kekz3z7luxolbhima 2844090 2844086 2026-11-03T03:45:57Z James500 297601 /* Fashion */ Add 2844090 wikitext text/x-wiki {{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}} {{Bibliography}} {{research}} If this resource is ever completed, it will be a universal bibliography.<ref>See [[w:Bibliography]].</ref> Until then, it will be an approximation of a universal bibliography. This bibliography is arranged as an index of topics. ==Index== *[[Universal Bibliography/Bibliography|Bibliography]] *[[Universal Bibliography/Libraries|Libraries]] *[[Universal Bibliography/Literature|Literature]] *[[Universal Bibliography/Languages|Languages]] *[[Universal Bibliography/SF|SF]] *[[Universal Bibliography/Music|Music]] *[[Universal Bibliography/Cinema|Cinema]] *[[Universal Bibliography/Television|Television]] *[[Universal Bibliography/Culture|Culture]] *[[Universal Bibliography/Publishers and imprints|Publishers and imprints]] *[[Universal Bibliography/Printing|Printing]] *[[Universal Bibliography/Printers|Printers]] *[[Universal Bibliography/Microform|Microform]] *[[Universal Bibliography/Periodicals|Periodicals]] *[[Universal Bibliography/Reference|Reference]] *[[Universal Bibliography/Gazetteers|Gazetteers]] *[[Universal Bibliography/Humanities|Humanities]] *[[Universal Bibliography/Law|Law]] *[[Universal Bibliography/History|History]] *[[Universal Bibliography/Archaeology|Archaeology]] *[[Universal Bibliography/Geography|Geography]] *[[Universal Bibliography/Continents|Continents]] *[[Universal Bibliography/Countries|Countries]] *[[Universal Bibliography/Architecture|Architecture]] *[[Universal Bibliography/Mathematics|Mathematics]] *[[Universal Bibliography/Computers|Computers]] *[[Universal Bibliography/Kites|Kites]] *[[Universal Bibliography/Nostalgia|Nostalgia]] *[[Universal Bibliography/Children's non-fiction|Children's non-fiction]] ===About=== *[[Universal Bibliography/About|About]] ==Online libraries== Swedish: *[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank) *[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg) French *[[w:Persée (web portal)|Persée]] (persee.fr) Japanese *[[w:Aozora Bunko|Aozora Bunko]] (aozora.gr.jp) ==Biographical dictionaries etc== See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]] *Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Arthur, "Biographical Dictionaries in the Digital Era". Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&pg=PA83#v=onepage&q&f=false Page 83] et seq. Bibliographies, indexes, etc: *Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ] *Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ] *Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ] *Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ] *Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ] *Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ] *Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ] Particular works: *Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]] *Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&btnG=Search+Books&bksoutput=html_text&tbm=bks&tbo=1 editions:uzt3-qMuFcMC]) *A & C Black's Who's Who *Who Was Who *The Academic Who's Who. A & C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&pg=PA208&output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC *Dictionary of Edwardian Biography (Pike's New Century Series) Works that comprise largely of biographies: *The Penguin Companion to Literature Theatres *A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers & Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15] ==Dictionaries of dates== [https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org] *Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ] *Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&tbm=bks&q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC] *Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880 *[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ] *Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440] *Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930. *[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]] *Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates. *Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will] *Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA93&output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&pg=PA351&output=html_text] *Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May) *Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&id=k3dJAAAAYAAJ&output=html] *Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880] Australia *John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog] *John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford & Co. Brisbane. 1895. America *Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921. ==Commodity dictionaries== *Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&pg=PR22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&pg=RA15-PA7#v=onepage&q&f=false] *Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, "Commodity Dictionary", Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&pg=PA182#v=onepage&q&f=false p 182] & 183 *Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 & 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ] *Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC] *Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&pg=PA78#v=onepage&q&f=false] ==Encyclopedias== See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]] *[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]] *Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990. *The Dorling Kindersley Illustrated Family Encyclopedia ==Almanacs== See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]]. *Year Book and Almanac of Newfoundland. **For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft] *Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C] ==Censuses== *Census of New Zealand and Labrador **1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog] **1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng] **1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng] ==Pilot guides== *[[w:United States Coast Pilot|United States Coast Pilot]] *American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&pg=PR1#v=onepage&q&f=false] *Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ] *Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ] *Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&pg=PR1#v=onepage&q&f=false] *Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&pg=PP7#v=onepage&q&f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&pg=PP5#v=onepage&q&f=false] ==Books of facts== *The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC *Chambers Book of Facts *Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008 *Handy Book of Facts: Things Everyone Should Know. C.S. Hammond & Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ] ==Series of books== See [[w:Category:Series of books]] and [[w:Category:Monographic series]] *George M Sinkankas, "Series" in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&pg=PA250#v=onepage&q&f=false 250] to 273. *"Publishing in Series, 1896-1916" in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&pg=PA539#v=onepage&q&f=false Page 539] et seq. *Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&pg=PP1#v=onepage&q&f=false vol 2]. *Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ] *Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Khan. "Monographs in series". The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&pg=PA208#v=onepage&q&f=false 207] to 209. *Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&pg=PA6#v=onepage&q&f=false 6] and 7. *Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&pg=PA11#v=onepage&q&f=false] **Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&pg=PA1190#v=onepage&q&f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&pg=PA53#v=onepage&q&f=false] ***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ] **Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ] **Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ] **Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ] *Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ] **2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ] **3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA63#v=onepage&q&f=false] *Ocran, Emmanuel Benjamin. Scientific & Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&pg=PA949#v=onepage&q&f=false] *Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ] *Young People's Literature in Series *Catalog of Reprints in Series. (sometimes called "Catalogue of Reprints in Series"). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA73#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ] *Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&pg=PA656#v=onepage&q&f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&pg=PA367#v=onepage&q&f=false] ==Series of classics== *Penguin Classics (Penguin Modern Classics, Penguin English Library) *Oxford World Classics *Everyman's Library *Wordsworth Classics *Macmillan Collectors Library *Bantam Classics *Minster Classics *The Literary Heritage Collection (Heron Books, London. William Collins Sons & Co, Glasgow) *Chandos Classics *Temple Classics *Longmans Heritage of Literature Series Russian *Greatest Masterpieces of Russian Literature (Heron Books, London) SF *Corgi SF Collectors Library Children's and shorter classics etc *Shorter Classics. Ginn and Company. *Ladybird Children's Classics. *Mini Classics. Parragon Books. *Bonny Books. Peter Haddock Ltd. *A series published by Dean & Son Ltd ==Non-fiction general series== *[[w:Oxford Companions|Oxford Companions]] *[[w:Cambridge Companions|Cambridge Companions]] *Princeton Companions *Blackwell Companions. Wiley Blackwell Companions *Routledge Companions. Routledge Research Companions *Ashgate Companions. Ashgate Research Companions *Brill's Companions *Facts on File Companions *Guides to Information Sources. Bowker-Saur *Butterworths Guides to Information Sources. *Columbia Guides *Blackwell Guides *Edinburgh Critical Guides *Collins Reference Dictionaries *New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]]) *Collins Gem (see [[w:List of Collins GEM books]]) *Concise Encyclopedias. Collins. *Time Life Books (see [[w:Time Life#Book series]]) *[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. *[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton. *Made Simple Books. W H Allen. *Palgrave Master Series *Harrap's Mini Series *Shire Albums. Shire Publications. *Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd. *The Wonderful World Books. Macdonald and Company *Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners. *Hamlyn Pocket Guides *Oxford Monograph Series *Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]] *Helpmate Handbooks. Willow Books University *University Paperbacks. Meuthen & Co *World Student Series. Addison Wesley *Unibooks. Hodder and Stoughton *International Student Editions. Van Nostrand Reinhold *Hutchinson University Library Imprints *Pelican Books Pictorials *Salmon Cameracolour series *Pitkin Pictorials United Kingdom *Aspects of Britain. HMSO. Places *The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]] *G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]] Art *Movements in World Art. Meuthen. *Movements in Modern Art. Meuthen. *How to Draw and Paint. New Burlington. Film *BFI Companions Popular science *Contemporary Science Paperbacks. Oliver and Boyd. *Pan Piper Science Series Science and mathematics *Simon and Schuster Tech Outlines *Schaum's Outline Series Military *Illustrated Military Guides. Illustrated Guides. "An Illustrated Guide to ...". Salamander Books. *Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press. *Osprey Men-at-Arms *Jane's Pocket Books Communication *The Library of Communication Techniques. Focal Press. *John Fiske (ed). Studies in Culture and Communication. Routledge. *The Media. Wayland. Cookery *ABC series. Peter Pauper Press. Gardening *Pan Piper Small Gardens Series. Mythology *Series on mythology published by Southwater (imprint of Anness) ==History and Geography== See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]]. *Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&pg=PP1#v=onepage&q&f=false] *Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&pg=PP1#v=onepage&q&f=false] General series *Cambridge Studies in Historical Geography Atlases *The Times Atlas of World History *Philip's Atlas of World History History of geography: *Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ] ==Chronology== See also [[Universal Bibliography/History#Millennia, centuries and decades]] General *Chronology of World History. **Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994. **Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994. **Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994. **Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon & Schuster. Reissued 1994. Centuries *Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ] *Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ] *Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC] *20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ] *Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ] *Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&pg=PP7#v=onepage&q&f=false] Decades *Series: **Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ] Years *Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&pg=PA113#v=onepage&q&f=false Catalogue]. *The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&pg=PP7#v=onepage&q&f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&pg=PP7#v=onepage&q&f=false 1882 to 1883]. *James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&pg=PP7#v=onepage&q&f=false] *[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&pg=PR1#v=onepage&q&f=false 1821]. *Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&pg=PR3#v=onepage&q&f=false] Ancient *Bickerman. Chronology of the Ancient World. 1968. *Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993. ==Anniversaries== *Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ] *On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&pg=PA1#v=onepage&q&f=false] ==Egyptology== *Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&pg=PR3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&pg=PR3#v=onepage&q&f=false] ==Battlefields== *[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series] *"The Times Guide to Battlefields of Britain". Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart. ==Armed forces== Periodicals: *[[w:NATO Review|NATO Review]] Military *The Journal of Military History *Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC] *The Monthly Army List [Google editions:I0t2L4ElznEC] *The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC] *Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC] *The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&q=editions:dEauCcI7kssC&biw=534&bih=736&dpr=1.5#sbfbu=1] *The Royal Tank Corps Journal *The Tank [https://www.google.com/search?sa=N&cs=0&tbm=bks&q=editions:Dv-RbpoM7acC&biw=534&bih=736&dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment *The Cavalry Journal [https://www.google.com/search?sa=N&cs=0&tbm=bks&q=editions:cVQlfkRl6KUC&biw=534&bih=688&dpr=1.5#sbfbu=1] *The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&q=editions:liFy4uc0ggYC&biw=534&bih=736&dpr=1.5] *Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC] *The Royal Engineers Journal [https://www.google.com/search?tbm=bks&q=editions:8XobinXLbD0C&biw=534&bih=736&dpr=1.5] *Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C] *Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC] United States *Military Review *The Coast Artillery Journal [Google editions:nMCogSJ_rlkC] *Infantry Journal [Google editions:ULqoLmbUR5cC] *The Reserve Officer [Google editions:JQDRDrnD1QQC] Naval *[[w:Navy News|Navy News]] ==Armour== Armoured warfare; tank warfare *Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ] *Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ] *Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ] *Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ] *Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ] *Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&pg=PP1#v=onepage&q&f=false] *Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ] Atlas *Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ] Armored forces *Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ] ==Mesoamerica== *James. Aztecs & Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: "Books Received" (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and "Overviews for the general reader" (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252. *Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ] ==Accounting== See [[s:Category:Accounting]] Periodicals *[[s:The Accountant|The Accountant]] (1874 onwards) *Accountant's Magazine (1897 onwards) Aberdeen ==Arts== *Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997. ==Biography== *Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&pg=PP1#v=onepage&q&f=false] *Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Periodicals *Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ] *Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ] Yearbooks *Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC] *Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ] ==Information technology== *Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&pg=PR1#v=onepage&q&f=false] ==Economics== General series: *Dryden Press Series in Economics *Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational *Nuffield Economics & Business. Nuffield Foundation. Longman. Other: *Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984. *Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991. *Anderton, Alain. Economics. Causeway Press. 1991. *Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983. *Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991. *Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986. *Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989. *Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985.  *Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981. *National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990. *Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992. ==Games== Chess *Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987. *Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980. *Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977. *Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. General series *Batsford Chess Books **Discovering Chess Series. B T Batsford. Periodicals See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]] *British Chess Magazine Othello *Goro Hasegawa with Maxine Brady. How to Win at Othello. (A Harvest/HBJ book). Jove Publications. 1977. ISBN 0-15-642215-8. [https://books.google.co.uk/books?id=mTdhlwVeyCcC] Wargames *Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward. **Laurie Taylor. "Attila the Hun invades Tyne Tees". TV Times. 1978. pp 28 & 29. **Terry Wise. "Battleground". Battle for Wargamers. June 1978. pp 261 & 262. *[[w:Game of War|Game of War]]. Channel 4. 1997. [Television]. ==Toys== Periodicals *Games & Toys: The Leading Trade Journal for Home & Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C *Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. "Harcourt To Buy Journals From Haire Publishing Co" [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29 *[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. "Playthings bought by Geyer-McAllister" [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing Capsule toys Gachapon *Tim Hornyak. [https://www.japantimes.co.jp/life/2017/08/19/lifestyle/gachapon-tracing-evolution-japans-colorful-toy-capsules/ Gachapon: Tracing the evolution of Japan’s colorful toy capsules]. The Japan Times. 19 August 2017. ==Cricket== See [[w:Bibliography of cricket]] *Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ]. **Peter Arnold. The Illustrated Encyclopedia of World Cricket. *Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ] Scores and biographies *Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&pg=PR3#v=onepage&q&f=false] **See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]] Periodicals *[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&pg=PP7#v=onepage&q&f=false]. Australia *Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463] *The Oxford Companion to Australian Cricket India *The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies] Annuals *[[w:Indian Cricket (annual)|Indian Cricket]]. [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966]. ===Cricketers=== Cricketers, including biographical dictionaries and collections of biographies *[[w:ESPNcricinfo|ESPNcricinfo]] *[[w:CricketArchive|CricketArchive]] *John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ] *World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ] *Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ] *Sproat. Debrett's Cricketers' Who's Who. 1980. *S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934. *Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ] *Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ] *Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC] *Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ] Gloucestershire *Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html] *Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C]. *Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795] Australia *The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ] *Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ] *C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894. *Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus & Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ]. ==Geology== *Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History. ==Mineralogy== *Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&pg=PA473#v=onepage&q&f=false 473] to 476. *Battey, Maurice Hugh. Mineralogy for students. Oliver & Boyd. 1972. 2nd Ed. Longman. 1981. ==Paper== See [[s:Category:Paper]] *Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf] *West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog] ==Books== *British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ] *Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ] *Book News. 1882 to 1918. (John Wanamaker). Called "Book News Monthly" from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&pg=PP7#v=onepage&q&f=false] *Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ] *U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ] *Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ] *Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ] *Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C] *Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ] Rare *Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ] Printed *Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&pg=PP1#v=onepage&q&f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&pg=PP1#v=onepage&q&f=false vol 31] ==Paperback and Paperbound== *Swados, "Paper Books: What do they Promise?" (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114 *Wagman, "The Paperbound Book Business" (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&pg=RA4-PA9#v=onepage&q&f=false 9] (No 5, November) ==Science== *Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998. ==Entertainment== *The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/] ==Fiction== ===Characters=== *Bowker's Guide to Characters in Fiction [https://books.google.co.uk/books?id=rVjlAAAAMAAJ] *William Freeman. Dictionary of Fictional Characters. [https://books.google.co.uk/books?id=sRYuAAAAMAAJ] *Martin Seymour-Smith. Dictionary of Fictional Characters. The Writer, Inc. [https://books.google.co.uk/books?id=MoQYAAAAIAAJ] *Cyclopedia of Literary Characters. Salem Press. Rev Ed: 1998. [https://books.google.co.uk/books?id=IJMYAAAAIAAJ] *Larousse Dictionary of Literary Characters. [https://books.google.co.uk/books?id=hq4UAQAAIAAJ] *Chambers Dictionary of Literary Characters. [https://books.google.com/books?id=KWNMPwAACAAJ] *Michael D Sollars. Dictionary of Literary Characters. [https://books.google.co.uk/books?id=IJddAQAACAAJ] ==Comedy and tragedy== *E H Mikhail. Comedy and Tragedy: A Bibliography of Critical Studies. 1972. [https://books.google.co.uk/books?id=5bgEAAAAMAAJ] ===Comedy=== *James E Evans. Comedy: An Annotated Bibliography of Theory and Criticism. 1987. [https://books.google.co.uk/books?id=s9ZgthAs1t8C&pg=PP1#v=onepage&q&f=false] *Meghan Duffy (ed). Daniel Gerould (senior ed). Comedy: A Bibliography of Critical Studies in English on the Theory and Practice of Comedy in Drama, Theatre, and Performance. New York. 2006. [https://books.google.co.uk/books?id=fI0jAQAAIAAJ] *David Humphrey. The Time of Laughter: Comedy and the Media Cultures of Japan. University of Michigan Press. 2023. [https://books.google.co.uk/books?id=CkTLEAAAQBAJ&pg=PP1#v=onepage&q&f=false] ==Radio and television== *William E McCavitt. Radio and Television: A Selected Annotated Bibliography. Scarecrow Press. 1978. [https://books.google.com/books?id=DAQVAQAAIAAJ] *Gertrude G Broderick. Radio and Television Bibliography. Bulletin 1948 No 17. [https://books.google.co.uk/books?id=yIPP1dsmNtAC&pg=PP5#v=onepage&q&f=false] *Patricia Beall Hamill. Radio and Television: A Selected Bibliography. Bulletin 1960 No 25. [https://books.google.co.uk/books?id=AJhGAQAAMAAJ&pg=PP3#v=onepage&q&f=false] ==Film and television== *Jean Mitry. Bibliographie internationale du cinéma et de la télévision. 1966 onwards. [https://books.google.co.uk/books?id=SC3gAAAAMAAJ] Commentary: A Reference Guide for English Studies, [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA439#v=onepage&q&f=false p 439]. ==Broadcasting== *Iwao Nakajima. The Broadcasting Industry in Japan: Its Historical, Legal and Economic Aspects. (International Studies of Broadcasting). NHK Radio and TV Culture Research Institute. 1971. [https://books.google.co.uk/books?id=uZ4vAQAAIAAJ] *50 Years of Japanese Broadcasting. NHK. 1977. [https://books.google.com/books?id=FoSKqY7tG-AC] *The History of Broadcasting in Japan. Nippon Hoso Kyokai. 1967. [https://books.google.com/books?id=I1IHAQAAIAAJ] *Hitoshi Mitomo and Mikio Kimura. Broadcasting in Japan: Challenges and Opportunities. [https://books.google.co.uk/books?id=7hCcEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Broadcasting in Japan: Case-studies on Broadcasting Systems. 1978. 2011. [https://books.google.co.uk/books?id=Ez5ZBwAAQBAJ&pg=PR4#v=onepage&q&f=false] *"World's First TV Stereo Broadcasts in Japan". Japan Report. (vol 25, no 3). 1 February 1979. p 4. *Broadcasting in Japan: A Historical Review [https://books.google.com/books?id=uDYlAAAAMAAJ] Annuals *[[w:ja:日本民間放送年鑑|日本民間放送年鑑]] (Japan Commercial Broadcasting Yearbook). コーケン出版. [https://books.google.co.uk/books?id=vh22AAAAIAAJ 1997]. ==Animation== *John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005. *Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&pg=PA1958#v=onepage&q&f=false]. *[https://www.bloomberg.com/news/articles/2026-10-07/anime-s-popularity-is-rising-among-teen-fans Anime is winning over a new generation of fans]. Bloomberg. 9 October 2026. [https://www.japantimes.co.jp/culture/2026/10/09/entertainment-news/anime-survey-watching/ The Japan Times]. ==Colours== *Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/] ==Bilateral== Britain and Japan *Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ] ==Prehistoric life== Prehistoric animals *[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.] *Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&pg=PA269#v=onepage&q&f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&pg=PA5#v=onepage&q&f=false]. **Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977. Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ] **Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme. Dinosaurs *Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.] ==Regions and other places== *JD Pearson. Oriental and Asian Bibliography: An Introduction with Some Reference to Africa. Archon Books. [https://books.google.com/books?id=P18ZAAAAMAAJ] *Theodore Besterman. A World Bibliography of Oriental Bibliographies. Revised by JD Pearson. Rowman and Littlefield. [https://books.google.co.uk/books?id=dvnfAAAAMAAJ] ==Culture and society== *Victoria Lyon Bestor, Theodore C Bestor and Akiko Yamagata (eds). Routledge Handbook of Japanese Culture and Society. 2011. [https://books.google.co.uk/books?id=0cBYffHp5L4C&pg=PP1#v=onepage&q&f=false] *Josef Kreiner and Hans-Dieter Ölschleger (eds). Japanese Culture and Society: Models of Interpretation. 1996. [https://books.google.co.uk/books?id=6skwAQAAIAAJ] *Dolores P Martinez. Modern Japanese Culture and Society. 2007. [https://books.google.co.uk/books?id=l6kMAQAAMAAJ vol 1] *C Andrew Gerstle. 18th Century Japan: Culture and Society. 1989. [https://books.google.co.uk/books?id=ViglDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Periodicals *Review of Japanese Culture and Society [https://books.google.co.uk/books?id=W_o-AQAAIAAJ] ==Clothes== *John Marshall. Make Your Own Japanese Clothes: Patterns and Ideas for Modern Wear. Kodansha International. 1988. [https://books.google.co.uk/books?id=ljOuuy0OO0UC&pg=PP1#v=onepage&q&f=false] ==Fashion== *Toby Slade. Japanese Fashion: A Cultural History. 2009. [https://books.google.co.uk/books?id=AInI19E48QIC&pg=PP3#v=onepage&q&f=false] *Bonnie English. Japanese Fashion Designers: The Work and Influence of Issey Miyake, Yohji Yamamotom, and Rei Kawakubo. 2011. [https://books.google.co.uk/books?id=6UMfAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sadao Hibi. Japanese Detail Fashion. Chronicle Books. 1989. [https://books.google.com/books?id=G-kIAQAAMAAJ] **Nihon No Dento Iro To Katachi. Graphic-sha Publishing. 1987. Kimono *Sheila Cliffe. The Social Life of Kimono: Japanese Fashion Past and Present. 2017. [https://books.google.co.uk/books?id=7G6xDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Anna Jackson (ed). Kimono: The Art and Evolution of Japanese Fashion. V&A Publishing. 2010. [https://books.google.co.uk/books?id=8gtMAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/kimono-the-art-and-evolution-of-japanese-fashion-the-khalili-collections/oclc/917375803]. ==Documents== *Pages from History: Ten Documents that changed the World AD 1000 to AD 2000. The Sunday Times. (10 pull out supplements and folder). *Scott Christianson. 100 Documents That Changed the World: From Magna Carta to WikiLeaks. [https://books.google.co.uk/books?id=rxiUEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Joseph Janes. Documents That Changed the Way We Live. 2017. [https://books.google.co.uk/books?id=a6-zDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Graham Stewart. Britannia: 100 Documents that Shaped a Nation. [https://books.google.co.uk/books?id=XUuL3xyZ9k8C&pg=PA1#v=onepage&q&f=false] ==See also== *[[Bibliography]] ==Notes== {{Reflist}} {{subpagesif}} [[Category:Bibliographies]] [[Category:Research]] q60oj4t49tmyxdtdha79nhj6hivfjzk 2844150 2844090 2026-11-03T09:16:11Z James500 297601 /* Fashion */ Add 2844150 wikitext text/x-wiki {{Center top}}{{Resize|3em|'''Bibliotheca Universalis'''}}{{Center bottom}} {{Bibliography}} {{research}} If this resource is ever completed, it will be a universal bibliography.<ref>See [[w:Bibliography]].</ref> Until then, it will be an approximation of a universal bibliography. This bibliography is arranged as an index of topics. ==Index== *[[Universal Bibliography/Bibliography|Bibliography]] *[[Universal Bibliography/Libraries|Libraries]] *[[Universal Bibliography/Literature|Literature]] *[[Universal Bibliography/Languages|Languages]] *[[Universal Bibliography/SF|SF]] *[[Universal Bibliography/Music|Music]] *[[Universal Bibliography/Cinema|Cinema]] *[[Universal Bibliography/Television|Television]] *[[Universal Bibliography/Culture|Culture]] *[[Universal Bibliography/Publishers and imprints|Publishers and imprints]] *[[Universal Bibliography/Printing|Printing]] *[[Universal Bibliography/Printers|Printers]] *[[Universal Bibliography/Microform|Microform]] *[[Universal Bibliography/Periodicals|Periodicals]] *[[Universal Bibliography/Reference|Reference]] *[[Universal Bibliography/Gazetteers|Gazetteers]] *[[Universal Bibliography/Humanities|Humanities]] *[[Universal Bibliography/Law|Law]] *[[Universal Bibliography/History|History]] *[[Universal Bibliography/Archaeology|Archaeology]] *[[Universal Bibliography/Geography|Geography]] *[[Universal Bibliography/Continents|Continents]] *[[Universal Bibliography/Countries|Countries]] *[[Universal Bibliography/Architecture|Architecture]] *[[Universal Bibliography/Mathematics|Mathematics]] *[[Universal Bibliography/Computers|Computers]] *[[Universal Bibliography/Kites|Kites]] *[[Universal Bibliography/Nostalgia|Nostalgia]] *[[Universal Bibliography/Children's non-fiction|Children's non-fiction]] ===About=== *[[Universal Bibliography/About|About]] ==Online libraries== Swedish: *[[w:Swedish Literature Bank|Litteraturbanken]] (Swedish Literature Bank) *[[w:Project Runeberg|Projekt Runeberg]] (Project Runeberg) French *[[w:Persée (web portal)|Persée]] (persee.fr) Japanese *[[w:Aozora Bunko|Aozora Bunko]] (aozora.gr.jp) ==Biographical dictionaries etc== See [[w:Bibliography of encyclopedias: general biographies]] and [[w:List of biographical dictionaries]] *Fox. 'True Biographies of Nations?': The Cultural Journeys of Dictionaries of National Biography. ANU Press. 2019 [https://books.google.co.uk/books?id=siSbDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Arthur, "Biographical Dictionaries in the Digital Era". Advancing Digital Humanities: Research, Methods, Theories. 2014. Chapter 6. [https://books.google.co.uk/books?id=z7MaBgAAQBAJ&pg=PA83#v=onepage&q&f=false Page 83] et seq. Bibliographies, indexes, etc: *Wynar. ARBA Guide to Biographical Dictionaries. Libraries Unlimited. 1986 [https://books.google.co.uk/books?id=5FfgAAAAMAAJ] *Slocum, Robert B (ed). Biographical Dictionaries and Related Works. Gale Research Company. 2nd Ed: 1986 [https://books.google.co.uk/books?id=5uMpAQAAMAAJ] *Biographical Dictionaries Master Index. (Gale Biographical Index Series). [https://books.google.co.uk/books?id=ZEshAQAAMAAJ] [https://books.google.co.uk/books?id=pPAzAQAAIAAJ] see also [https://books.google.co.uk/books?id=o_gPAQAAMAAJ] *Children's Authors and Illustrators: An Index to Biographical Dictionaries. (Gale Biographical Index Series). 2nd Ed: 1978,  3rd Ed: 1981, 4th Ed: 1987 [https://books.google.co.uk/books?id=VIsWAQAAMAAJ] [https://books.google.co.uk/books?id=DFtGAQAAIAAJ] [https://books.google.co.uk/books?id=01wjAQAAIAAJ] *Index to the Wilson Authors Series [https://books.google.co.uk/books?id=oNZkAAAAMAAJ] *Auchterlonie. Arabic Biographical Dictionaries: A Summary Guide and Bibliography. 1987 [https://books.google.co.uk/books?id=rW59QgAACAAJ] *Black Biographical Dictionaries, 1790-1950 [https://books.google.co.uk/books?id=laIUAQAAMAAJ] Particular works: *Oxford Dictionary of National Biography; [[w:Dictionary of National Biography|Dictionary of National Biography]] *Boase. Modern English Biography. ([http://www.google.com/search?q=editions%3Auzt3-qMuFcMC&btnG=Search+Books&bksoutput=html_text&tbm=bks&tbo=1 editions:uzt3-qMuFcMC]) *A & C Black's Who's Who *Who Was Who *The Academic Who's Who. A & C Black. 1st Ed: 1973 [https://books.google.co.uk/books?id=dnUWAQAAMAAJ] [https://books.google.co.uk/books?id=fXJmAAAAMAAJ]. 2nd Ed: 1975. Commentary: [https://books.google.co.uk/books?id=7VyOANl2qxoC&pg=PA208&output=html_text]. GBooks: editions:INAP7GGD2gYC editions:tA0FkHC75FIC *Dictionary of Edwardian Biography (Pike's New Century Series) Works that comprise largely of biographies: *The Penguin Companion to Literature Theatres *A Biographical Dictionary of Actors, Actresses, Musicians, Dancers, Managers & Other Stage Personnel in London, 1660-1800. [https://books.google.co.uk/books?id=TGgS9VxWJ0oC vol 15] ==Dictionaries of dates== [https://archive.org/search.php?query=%22dictionary%20of%20dates%22 Archive.org] *Baxter Dictionary of Dates and Events. 1st Ed: 1963: Napier, M (ed). 2nd Ed: 1971: Sanders and Laffin. Commentary: 92 Library Journal 1819 [https://books.google.co.uk/books?id=CExVAAAAYAAJ] *Beeching, Cyril Leslie. A Dictionary of Dates. OUP. 1st Ed: 1993. 2nd Ed: 1997. [https://www.google.co.uk/search?hl=en&tbm=bks&q=editions:UGGp0EexZdcC editions:UGGp0EexZdcC] *Bolton, John. Bolton's Dictionary of Dates, arranged in alphabetical order. Foulsham. 1958. Review: [https://books.google.co.uk/books?id=awJPAAAAIAAJ 172] The Publisher 880 *[[w:William Darling (politician)|William Young Darling]]. A Book of Days: A Dictionary of Dates, a Chronology of Circumstance, the Face of Time. Richards Press. 1951. [https://books.google.co.uk/books?id=PLkfAAAAMAAJ] *Everyman's Dictionary of Dates. 1st Ed: 1911. 6th Ed: 1971. Review: (1971) 11 RQ 164 [http://www.jstor.org/stable/25824440] *Platt, Charles. Foulsham's Dictionary of Dates and General Information. 1930. *[[w:Haydn's Dictionary of Dates|Haydn's Dictionary of Dates]] *Hamlyn Dictionary of Dates and Anniversaries. Newnes Dictionary of Dates. *Williams, Henry Llewellyn. Hurst's Dictionary of Dates. 1891. [https://archive.org/details/hurstsdictionary00will] *Keller, Helen Rex. The Dictionary of Dates. Macmillan. 1934. Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA93&output=html_text] [https://books.google.co.uk/books?id=sAHfY6QbOEwC&pg=PA351&output=html_text] *Nelson's Dictionary of Dates. A Dictionary of Dates. (Nelson's Encyclopaedic Library). 1912 [https://books.google.co.uk/books?id=Mp9lvwEACAAJ]. Reviews: (June 1912) Journal of Education, vol 34 (New Series), vol 44 (Old Series), p 392 [https://books.google.co.uk/books?id=QIRFAQAAMAAJ]; (1912) [https://books.google.co.uk/books?id=9i4_AQAAIAAJ 108] The Spectator [http://archive.spectator.co.uk/article/18th-may-1912/25/a-dictionary-of-dates-vol-i-and-english-idioms-nel 805] (18 May) *Pulman, George Palmer. The World's Progress: A Dictionary of Dates. New York. 1861. [https://books.google.co.uk/books?printsec=frontcover&id=k3dJAAAAYAAJ&output=html] *Urdang, Laurence. The World Almanac Dictionary of Dates. Longman. 1982. [https://books.google.co.uk/books?id=I4IRAQAAMAAJ] Review: (1982) 22 RQ 101 [http://www.jstor.org/stable/25826880] Australia *John Henniker Heaton. Australian Dictionary of Dates and Men of the Time. 1879. [https://archive.org/details/australiandicti00heatgoog] *John James Knight. In the Early Days; History and Incident of Pioneer Queensland, with Dictionary of Dates in Chronological Order. Sapsford & Co. Brisbane. 1895. America *Damon, Charles Ripley. The American Dictionary of Dates, 458-1920. R G Badger. 1921. ==Commodity dictionaries== *Statistical Classification of Domestic and Foreign Commodities Exported from the United States. Commentary: [https://books.google.co.uk/books?id=91GLhsJSBj8C&pg=PR22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=RPwhAQAAMAAJ&pg=RA15-PA7#v=onepage&q&f=false] *Tovarnyi slovar'. (Commodity Dictionary). Reviews and commentary: Petrov, "Commodity Dictionary", Ekonomicheskaya Gazeta, No 13, 30 October 1961, p 45; CDSP , 13 December 1961, p 46; (1962) [https://books.google.co.uk/books?id=2vMRAAAAIAAJ 13] Current Digest of the Soviet Press 47; (1958) 15 Quarterly Journal of Current Acquisitions 210 [https://books.google.co.uk/books?id=ZcvozpZAfpEC] [https://books.google.co.uk/books?id=S47qEIfyCr0C]; Fitzpatrick, Stalinism: New Directions, [https://books.google.co.uk/books?id=rD5FzoKnTE0C&pg=PA182#v=onepage&q&f=false p 182] & 183 *Szilágyi. Commodity Dictionary in Five Languages. Budapest. Közgazdasági és Jogi Könyvkiadó (Publishing House for Economics and Law). 1963 or 1964. Commentary: Books from Hungary, vols 4-6, pp 26 & 40 [https://books.google.co.uk/books?id=6kMiAQAAMAAJ] *Dictionnaire des produits: appellations et caractéristiques des produits francais de consommation courante, 1960. Commentary: Walford (ed), Guide to Reference Material Supplement, 1963, p 106 [https://books.google.co.uk/books?id=ej-9pHGR67oC] *Chūgoku Shōhin Jiten. (Chinese commodity dictionary). Tokyo. 1960. [https://books.google.co.uk/books?id=Wc61lS0xj6AC&pg=PA78#v=onepage&q&f=false] ==Encyclopedias== See [[s:Category:Encyclopedias]], [[w:Bibliography of encyclopedias]] and [[w:Lists of encyclopedias]] *[[w:en:Encyclopædia Britannica Eleventh Edition|Encyclopædia Britannica Eleventh Edition]] *Paton, John (ed). Knowledge Encyclopedia: 1979, 1981, 1988. New Discovery Encyclopedia: 1990. *The Dorling Kindersley Illustrated Family Encyclopedia ==Almanacs== See [[s:Category:Almanacs]], [[s:Portal:Almanacs]], [[w:List of almanacs]], [[w:Category:Almanacs]]. *Year Book and Almanac of Newfoundland. **For 1896. 1895. [https://archive.org/details/yearbooknfld189600newfuoft] *Whiteley. On This Date: A Day-by-Day Listing of Holidays, Birthday and Historic Events, and Special Days, Weeks and Months. 2002. [https://books.google.co.uk/books?id=sKCfomKSa74C] ==Censuses== *Census of New Zealand and Labrador **1901 Census. Tables 2 and 3. 1903. [https://archive.org/details/censusnewfoundl00bondgoog] **1911 Census. Table 1. 1914. [https://archive.org/details/1911981911fnfldv11914eng] **1921 Census. Tables 4 and 5. 1923. [https://archive.org/details/1921981921fnfldv451923eng] ==Pilot guides== *[[w:United States Coast Pilot|United States Coast Pilot]] *American Coast Pilot [https://books.google.co.uk/books?id=8GoDAAAAYAAJ&pg=PR1#v=onepage&q&f=false] *Sailing Directions: Newfoundland. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=A77fAAAAMAAJ] *Newfoundland Pilot. Canadian Hydrographic Service. [https://books.google.co.uk/books?id=z7zfAAAAMAAJ] *Maxwell. The Newfoundland Pilot. Hydrographic Office, Admiralty. London. 1878. [https://books.google.co.uk/books?id=vS4BAAAAQAAJ&pg=PR1#v=onepage&q&f=false] *Newfoundland Pilot. HO No 73. Hydrographic Office. Governement Printing Office, Washington. 4th Ed: 1919: [https://books.google.co.uk/books?id=YGoDAAAAYAAJ&pg=PP7#v=onepage&q&f=false]. Sailing Directions for Newfoundland. 5th Ed: 1931: [https://books.google.co.uk/books?id=cMUiGo3JK9QC&pg=PP5#v=onepage&q&f=false] ==Books of facts== *The Reader's Digest Book of Facts. 1st Ed: 1985. Reprinted with amendments: 1987: [https://books.google.co.uk/books?id=B8PmM_5Zm1MC]. (Review: Library Journal, [https://books.google.co.uk/books?id=EPDgAAAAMAAJ v 9], p 102, 1 Dec 1987, [http://www.bookverdict.com/details.xqy?uri=Product-94667328910921.xml Book Verdict].) 3rd Revised Ed: 1995: [https://books.google.co.uk/books?id=E5YhAQAAIAAJ]. GBooks: editions:nnJlLybWxbIC *Chambers Book of Facts *Crystal, David (ed). Penguin Book of Facts. [https://books.google.co.uk/books?id=k0sZAQAAIAAJ 2004]. 2nd Ed: 2008 *Handy Book of Facts: Things Everyone Should Know. C.S. Hammond & Company. 1914. [https://books.google.co.uk/books?id=h5wRAAAAIAAJ] ==Series of books== See [[w:Category:Series of books]] and [[w:Category:Monographic series]] *George M Sinkankas, "Series" in Kent, Lancour and Daily (eds).  Encyclopedia of Library and Information Science. Volume 27. Marcel Dekker. 1979. Pages [https://books.google.co.uk/books?id=jU3fwyjqS5UC&pg=PA250#v=onepage&q&f=false 250] to 273. *"Publishing in Series, 1896-1916" in Eliot, Simon (ed). History of Oxford University Press. Louis,  Wm Roger (ed). Volume 3: 1896-1970. Oxford University Press. 2013. [https://books.google.co.uk/books?id=YbcJAgAAQBAJ&pg=PA539#v=onepage&q&f=false Page 539] et seq. *Spiers, John. The Culture of the Publisher’s Series. Palgrave Macmillan. 2011. [https://books.google.co.uk/books?id=ASaHDAAAQBAJ&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=XCl-DAAAQBAJ&pg=PP1#v=onepage&q&f=false vol 2]. *Spiers, John. Serious about Series: American 'Cheap' Libraries, British 'Railway' Libraries and Some Literary Series of the 1890's. 2007. [https://books.google.co.uk/books?id=1hRXAAAAYAAJ] [https://books.google.co.uk/books?id=AS4yQwAACAAJ] *Rooney, Paul Raphael. Railway Reading and Late-Victorian Literary Series. Routledge. 2018. [https://books.google.co.uk/books?id=uX5aDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Khan. "Monographs in series". The Principles and Practice of Library Science. 1996. Pages [https://books.google.co.uk/books?id=sAHfY6QbOEwC&pg=PA208#v=onepage&q&f=false 207] to 209. *Friskney. New Canadian Library: The Ross-McClelland Years, 1952-1978. Pages [https://books.google.co.uk/books?id=jHIjCCXBX9kC&pg=PA6#v=onepage&q&f=false 6] and 7. *Books in Series. R R Bowker Company. Commentary: [https://books.google.co.uk/books?id=uQe04OSlA7YC&pg=PA11#v=onepage&q&f=false] **Books in Series in the United States, 1966-1975. R R Bowker. 1977. Review: (1977) 14 Choice [https://books.google.co.uk/books?id=_e08AQAAIAAJ&pg=PA1190#v=onepage&q&f=false 1190] (No 8, November). Commentary: [https://books.google.co.uk/books?id=LYAhAAAAQBAJ&pg=PA53#v=onepage&q&f=false] ***Books in Series Supplement: A Supplement to Books in Series in the United States, 1966-1975. 1978. [https://books.google.co.uk/books?id=hOAaAQAAMAAJ] **Books in Series. 3rd Ed. 1980. [https://books.google.co.uk/books?id=d_kaAQAAMAAJ] **Books in Series, 1876-1949. R R Bowker Company. 1982. [https://books.google.co.uk/books?id=TngvAQAAIAAJ] [https://books.google.co.uk/books?id=iVIyAQAAMAAJ] [https://books.google.co.uk/books?id=R2AjAQAAIAAJ] **Books in Series, 1985-89. [https://books.google.co.uk/books?id=yEkxAQAAIAAJ] *Baer, Eleanora Agnes. Titles in Series: A Handbook for Librarians and Students. Scarecrow Press. Vol 1 (Books Published Prior to January 1953). 1953: [https://books.google.co.uk/books?id=GgAYAAAAMAAJ]. Vol 2 (Books Published Prior to January 1957). 1957: [https://books.google.co.uk/books?id=oqsXAAAAMAAJ] **2nd Ed: 1964. [https://books.google.co.uk/books?id=gWlAAAAAIAAJ Vol 1]. [https://books.google.co.uk/books?id=tWpAAAAAIAAJ Vol 2]. Supplement to the Second Edition. 1967: [https://books.google.co.uk/books?id=zGARAQAAMAAJ]. Second Supplement to the Second Edition. 1971: [https://books.google.co.uk/books?id=WwXhAAAAMAAJ] **3rd Ed: 1978. Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA63#v=onepage&q&f=false] *Ocran, Emmanuel Benjamin. Scientific & Technical Series: A Select Bibliography. 1973: [https://books.google.co.uk/books?id=oy0EAAAAMAAJ] Review: [https://books.google.co.uk/books?id=fTCw_DQH6zkC&pg=PA949#v=onepage&q&f=false] *Rosenberg and Nichols. Young People's Books in Series: Fiction and Non-fiction, 1975-1991. Libraries Unlimited. 1992. [https://books.google.co.uk/books?id=REHhAAAAMAAJ] *Young People's Literature in Series *Catalog of Reprints in Series. (sometimes called "Catalogue of Reprints in Series"). 1940 onwards. [https://books.google.co.uk/books?id=MSI4AAAAIAAJ] [https://books.google.co.uk/books?id=6n1EAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA73#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1RxuAAAAMAAJ] *Kuitert, Lisa. Het ene boek in vele delen. De Uitgave van Literaire Series in Nederland 1850-1900. Uitgeverij de Buitenkant. Amsterdam. 1993. Commentary: [https://books.google.co.uk/books?id=jSDnRo7YrWwC&pg=PA656#v=onepage&q&f=false] [https://books.google.co.uk/books?id=szBcAAAAMAAJ] [https://books.google.co.uk/books?id=SVcVAQAAIAAJ] [https://books.google.co.uk/books?id=R8Pfs146nUAC&pg=PA367#v=onepage&q&f=false] ==Series of classics== *Penguin Classics (Penguin Modern Classics, Penguin English Library) *Oxford World Classics *Everyman's Library *Wordsworth Classics *Macmillan Collectors Library *Bantam Classics *Minster Classics *The Literary Heritage Collection (Heron Books, London. William Collins Sons & Co, Glasgow) *Chandos Classics *Temple Classics *Longmans Heritage of Literature Series Russian *Greatest Masterpieces of Russian Literature (Heron Books, London) SF *Corgi SF Collectors Library Children's and shorter classics etc *Shorter Classics. Ginn and Company. *Ladybird Children's Classics. *Mini Classics. Parragon Books. *Bonny Books. Peter Haddock Ltd. *A series published by Dean & Son Ltd ==Non-fiction general series== *[[w:Oxford Companions|Oxford Companions]] *[[w:Cambridge Companions|Cambridge Companions]] *Princeton Companions *Blackwell Companions. Wiley Blackwell Companions *Routledge Companions. Routledge Research Companions *Ashgate Companions. Ashgate Research Companions *Brill's Companions *Facts on File Companions *Guides to Information Sources. Bowker-Saur *Butterworths Guides to Information Sources. *Columbia Guides *Blackwell Guides *Edinburgh Critical Guides *Collins Reference Dictionaries *New Horizons. Thames and Hudson. ([[w:Découvertes Gallimard|Découvertes Gallimard]]) *Collins Gem (see [[w:List of Collins GEM books]]) *Concise Encyclopedias. Collins. *Time Life Books (see [[w:Time Life#Book series]]) *[[w:Teach Yourself|Teach Yourself Books]]. English Universities Press. *[[w:Teach Yourself|Teach Yourself Books]]. Hodder and Stoughton. *Made Simple Books. W H Allen. *Palgrave Master Series *Harrap's Mini Series *Shire Albums. Shire Publications. *Fax Pax: Knowledge in a Nutshell. Fax Pax Ltd. *The Wonderful World Books. Macdonald and Company *Harper's ABC series. Includes A-B-C of Housekeeping, A-B-C of Electricity, A-B-C of Gardening and A-B-C of Manners. *Hamlyn Pocket Guides *Oxford Monograph Series *Study Outline Series. H W Wilson. [[s:Page:Russian Literature - A Study Outline.djvu/61|(wikisource)]] *Helpmate Handbooks. Willow Books University *University Paperbacks. Meuthen & Co *World Student Series. Addison Wesley *Unibooks. Hodder and Stoughton *International Student Editions. Van Nostrand Reinhold *Hutchinson University Library Imprints *Pelican Books Pictorials *Salmon Cameracolour series *Pitkin Pictorials United Kingdom *Aspects of Britain. HMSO. Places *The Little Guides. Meuthen [[s:Page:Cornwall (Salmon).djvu/336|(wikisource)]] *G.W.R. Series of Travel Books [[s:Page:The Cornwall coast.djvu/391|(wikisource)]] Art *Movements in World Art. Meuthen. *Movements in Modern Art. Meuthen. *How to Draw and Paint. New Burlington. Film *BFI Companions Popular science *Contemporary Science Paperbacks. Oliver and Boyd. *Pan Piper Science Series Science and mathematics *Simon and Schuster Tech Outlines *Schaum's Outline Series Military *Illustrated Military Guides. Illustrated Guides. "An Illustrated Guide to ...". Salamander Books. *Combat Arms. Arco Military Books. Salamander Books. Prentice Hall Press. *Osprey Men-at-Arms *Jane's Pocket Books Communication *The Library of Communication Techniques. Focal Press. *John Fiske (ed). Studies in Culture and Communication. Routledge. *The Media. Wayland. Cookery *ABC series. Peter Pauper Press. Gardening *Pan Piper Small Gardens Series. Mythology *Series on mythology published by Southwater (imprint of Anness) ==History and Geography== See also [[Universal Bibliography/History|History]] and [[Universal Bibliography/Geography|Geography]]. *Baker. Geography and History: Bridging the Divide. 2003. [https://books.google.co.uk/books?id=e8yf5JcefpAC&pg=PP1#v=onepage&q&f=false] *Darby. Relations of History and Geography: Studies in England, France and the United States. 2002. [https://books.google.co.uk/books?id=Vl4ZfpnP7NwC&pg=PP1#v=onepage&q&f=false] General series *Cambridge Studies in Historical Geography Atlases *The Times Atlas of World History *Philip's Atlas of World History History of geography: *Dunbar, Gary S. The History of Modern Geography: An Annotated Bibliography of Selected Works. Garland. 1985. [https://books.google.co.uk/books?id=FX4WAQAAIAAJ] ==Chronology== See also [[Universal Bibliography/History#Millennia, centuries and decades]] General *Chronology of World History. **Neville Williams. Chronology of the Modern World: 1763 to the present time. 1st Ed: 1966. (1763 to 1992). 2nd Ed: 1994. **Neville Williams. Chronology of the Expanding World 1492 to 1762. 1969. Reissued 1994. **Storey. Chronology of the Medieval World 800 to 1491. 1973. Reissued 1994. **Mellersh. Chronology of the Ancient World 10,000 BC to AD 799. Barrie and Jenkins. 1976. Helicon. Simon & Schuster. Reissued 1994. Centuries *Chronology of the 20th Century. Helicon. 1995. [https://books.google.com/books?id=pjsOAQAAMAAJ] *Brownstone and Franck. Timelines of the 20th Century. [https://books.google.com/books?id=IZ6SQgAACAAJ] *Beal. 20th Century Timeline. 1985. [https://books.google.com/books?id=cFrG7LBObGoC] *20th Century Day by Day [https://books.google.com/books?id=kyxaAAAAYAAJ] [https://books.google.com/books?id=WiOAAAAACAAJ] *Chronicle of the 20th Century [https://books.google.co.uk/books?id=pt3DYbnZO8sC] [https://books.google.co.uk/books?id=Gd1WPQAACAAJ] *Boyle. The Chronology of the Eighteenth and Nineteenth Centuries. 1826. [https://books.google.co.uk/books?id=wDENAAAAYAAJ&pg=PP7#v=onepage&q&f=false] Decades *Series: **Day by Day. Facts on File. [https://books.google.com/books?id=WfClvwEACAAJ] [https://books.google.com/books?id=CWNvQgAACAAJ] Years *Brown, D Kinnear. History of the Year. (1884 to 1885). [https://books.google.co.uk/books?id=DmRWAAAAYAAJ&pg=PA113#v=onepage&q&f=false Catalogue]. *The History of the Year: A Narrative of the Chief Events and Topics of Interest. [https://books.google.co.uk/books?id=ljgIAAAAQAAJ&pg=PP7#v=onepage&q&f=false 1881 to 1882]. [https://books.google.co.uk/books?id=1DgIAAAAQAAJ&pg=PP7#v=onepage&q&f=false 1882 to 1883]. *James Mason. The History of the Year 1876. [https://books.google.co.uk/books?id=6DoIAAAAQAAJ&pg=PP7#v=onepage&q&f=false] *[[w:The Annual Register|The Annual Register]]. [A View of the History Politics and Literature of the Year YYYY.] [https://books.google.co.uk/books?id=SrJNAAAAcAAJ&pg=PR1#v=onepage&q&f=false 1821]. *Giusto Traina. 428AD: An Ordinary Year at the End of the Roman Empire. [https://books.google.co.uk/books?id=gLumDwAAQBAJ&pg=PR3#v=onepage&q&f=false] Ancient *Bickerman. Chronology of the Ancient World. 1968. *Smithsonian Timelines of the Ancient World: A Visual Chronology from the Origins of Life. Dorling Kindersley. 1st American Ed: 1993. ==Anniversaries== *Sian Facer (ed). On this Day: The History of the World in 366 Days. Octopus Illustrated Publishing, London. Crescent Books, New York and Avenel. 1992: [https://books.google.com/books?id=SYGQgwHTuE0C]. Other: [https://books.google.co.uk/books?id=W687MAEACAAJ] [https://books.google.co.uk/books?id=7ujArQEACAAJ] *On this Day: A History of the World in 366 Days. DK. 2021. [https://books.google.co.uk/books?id=x4I5EAAAQBAJ&pg=PA1#v=onepage&q&f=false] ==Egyptology== *Annual Egyptological Bibliography [https://books.google.co.uk/books?id=8MoUAAAAIAAJ&pg=PR3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=-eUUAAAAIAAJ&pg=PR3#v=onepage&q&f=false] ==Battlefields== *[[w:War Walks|War Walks]]. BBC2. 1996 to 1997. [Television series] *"The Times Guide to Battlefields of Britain". Day 1: The Times, 1 August 1994, p 8. Day 2: The Times, 2 August 1994, p 8. Day 3: The Times, 3 August 1994, p 6. Day 4: The Times, 4 August 1994, p 9. Day 5: The Times, 5 August 1994, p 9. Day 6: The Times, 6 August 1994, p 6. There was also a colour wall chart. ==Armed forces== Periodicals: *[[w:NATO Review|NATO Review]] Military *The Journal of Military History *Journal of the Royal United Service Institution [Google editions:lMJAgUvBWAEC editions:dcFNqS8JFjoC] *The Monthly Army List [Google editions:I0t2L4ElznEC] *The Army Quarterly and Defence Journal [Google editions:c7UjQ-q7SbUC] *Journal of the Society for Army Historical Research [Google editions:9HZkbMTl6mcC] *The Royal Armoured Corps Journal [https://www.google.com/search?tbm=bks&q=editions:dEauCcI7kssC&biw=534&bih=736&dpr=1.5#sbfbu=1] *The Royal Tank Corps Journal *The Tank [https://www.google.com/search?sa=N&cs=0&tbm=bks&q=editions:Dv-RbpoM7acC&biw=534&bih=736&dpr=1.5#ip=1] Editorial office at the Royal Tank Regiment *The Cavalry Journal [https://www.google.com/search?sa=N&cs=0&tbm=bks&q=editions:cVQlfkRl6KUC&biw=534&bih=688&dpr=1.5#sbfbu=1] *The Journal of the Royal Artillery [https://www.google.com/search?tbm=bks&q=editions:liFy4uc0ggYC&biw=534&bih=736&dpr=1.5] *Minutes of Proceedings of the Royal Artillery Institution [Google editions:wdjZ588FbtMC] *The Royal Engineers Journal [https://www.google.com/search?tbm=bks&q=editions:8XobinXLbD0C&biw=534&bih=736&dpr=1.5] *Journal of the Royal Electrical and Mechanical Engineers [https://books.google.com/books?id=dz0cmA1jnv4C] *Journal of the Royal Army Medical Corps [Google editions:FyUJx2dEWcQC] United States *Military Review *The Coast Artillery Journal [Google editions:nMCogSJ_rlkC] *Infantry Journal [Google editions:ULqoLmbUR5cC] *The Reserve Officer [Google editions:JQDRDrnD1QQC] Naval *[[w:Navy News|Navy News]] ==Armour== Armoured warfare; tank warfare *Harris and Toase. Armoured Warfare. 1990. [https://books.google.com/books?id=KYPfAAAAMAAJ] *Carver. The Apostles of Mobility: The Theory and Practice of Armoured Warfare. 1979. [https://books.google.com/books?id=8qcgAAAAMAAJ] *Fuller. Armoured Warfare: An Annotated Edition of Fifteen Lectures on Operations between Mechanized Forces. 1943. [https://books.google.co.uk/books?id=2E4tAQAAMAAJ] *Black. Tank Warfare. 2020. [https://books.google.co.uk/books?id=oFP5DwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Jorgensen and Mann. Tank Warfare. 2001. [https://books.google.co.uk/books?id=0AghAQAAIAAJ] *Searle. Armoured Warfare: A Military, Political and Global History. 2017. [https://books.google.co.uk/books?id=HN4CDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Willey. Tanks: The History of Armoured Warfare. 2018. [https://books.google.com/books?id=AXTltAEACAAJ] *Perrett. Iron Fist: Classic Armoured Warfare Case Studies. [https://books.google.co.uk/books?id=pKGyeWqJcCEC]. Iron Fist: Classic Armoured Warfare. [https://books.google.co.uk/books?id=KKcKI4dG0VUC&pg=PP1#v=onepage&q&f=false] *Tom Clancy. Armoured Warfare: Guided Tour of an Armoured Cavalry Regiment. [https://books.google.co.uk/books?id=UxhONAAACAAJ] Atlas *Stephen Hart (ed). Atlas of Armored Warfare: From 1916 to the Present Day. Metro Books. 2012. [https://search.worldcat.org/title/1391166759]. Atlas of Tank Warfare. [https://books.google.com/books?id=KWqppwAACAAJ] Armored forces *Ogorkiewicz. Armoured Forces: A History of Armoured Forces and Their Vehicles. 1970. [https://books.google.co.uk/books?id=qIHfAAAAMAAJ] ==Mesoamerica== *James. Aztecs & Maya: The Ancient Peoples of Middle America. Tempus. 2001. 2005. History Press. [https://books.google.co.uk/books?id=XOXNhTY6TCYC 2009]. Reviews: "Books Received" (2003) [https://books.google.co.uk/books?id=3dozAQAAIAAJ 14] Minerva 57 (No 1); and "Overviews for the general reader" (2002) [https://books.google.co.uk/books?id=qShmAAAAMAAJ 76] Antiquity 252. *Weaver. The Aztecs, Maya, and Their Predecessors. 1972. 2nd Ed: 1981: [https://books.google.co.uk/books?id=0mQkAQAAIAAJ] [https://books.google.com/books?id=OWQkAQAAIAAJ] ==Accounting== See [[s:Category:Accounting]] Periodicals *[[s:The Accountant|The Accountant]] (1874 onwards) *Accountant's Magazine (1897 onwards) Aberdeen ==Arts== *Murray (ed).The Hutchinson Dictionary of the Arts. Helicon Publishing. 1994. Paperback Ed: 1995. Reprinted 1997. ==Biography== *Parke. Biography: Writing Lives. 2002 [https://books.google.co.uk/books?id=6bAz2K98MeYC&pg=PP1#v=onepage&q&f=false] *Caine. Biography and History. (Theory and History). 1st Ed: 2010, 2nd Ed: 2019 [https://books.google.co.uk/books?id=h3dvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Periodicals *Biography. Biography: An Interdisciplinary Quarterly. 1978 onwards. Published by the University Press of Hawaii for the Biographical Research Center. [https://books.google.co.uk/books?id=s84ZAAAAYAAJ] *Biography News. 1974 to 1975. Gale Research Company. [https://books.google.co.uk/books?id=RRsXAQAAIAAJ] Yearbooks *Current Biography Yearbook [https://books.google.com/books?id=Zcml63jalMIC] *Dictionary of Literary Biography Yearbook [https://books.google.com/books?id=gNNlAAAAMAAJ] ==Information technology== *Haynes, David (ed). Information Sources in Information Technology. (Guides to Information Sources). Bowker Saur. 1990. [https://books.google.co.uk/books?id=0hYjAAAAQBAJ&pg=PR1#v=onepage&q&f=false] ==Economics== General series: *Dryden Press Series in Economics *Hurl, Bryan (ed). Studies in the UK Economy. Heinemann Educational *Nuffield Economics & Business. Nuffield Foundation. Longman. Other: *Bannock, Baxter and Davis. The Penguin Dictionary of Economics. Penguin Books. 4th Ed: 1987. Bannock, Baxter and Rees. 1972. 2nd Ed: 1978. 3rd Ed: 1984. *Begg, Fischer and Dornbusch. Economics. McGraw Hill. 1984. 2nd Ed: 1987. 3rd Ed: 1991. *Anderton, Alain. Economics. Causeway Press. 1991. *Maile, Roger. Economics. (Core Business Studies). Mitchell Beazly. 1983. *Maunder, Myers, Wall and Miller. Economics Explained. Collins Educational. 1987. 2nd Ed: 1991. *Tibbitt, Andrew. A guide to A Level Economics. Thomas Nelson and Sons. 1986. *Lipsey, Richard G. An Introduction to Positive Economics. Weidenfeld and Nicolson. 1963. 2nd Ed: 1966. 3rd Ed: 1971. 4th Ed: 1975. 5th Ed: 1979. 6th Ed: 1983. 7th Ed: 1989. *Nicolson, Walter. Microeconomic Theory: Basic Principles and Extensions. (Dryden Press Series in Economics). Dryden Press, Holt-Saunders. 3rd Ed: 1985.  *Caves and Jones. World Trade and Payments: An Introduction. Little, Brown and Company. 1973. 1977. 3rd Ed: 1981. *National Institute of Economic and Social Research. The UK economy. (Studies in the UK Economy). Heinemann Educational. 1990. *Smith, Charles. UK trade and sterling. (Studies in the UK Economy). Heinemann Educational. 1992. ==Games== Chess *Hooper and Whyld. The Oxford Companion to Chess. Oxford University Press. 1984. Paperback: 1987. *Golombek, Harry. The Game of Chess. 1954. 2nd Ed: 1963. 3rd Ed: 1980. *Pritchard, D. Brine. The Right Way to Play Chess. 1950. 8th Ed: 1971. 10th Ed: 1974. 11th Ed: 1977. *Horowitz, Al. From Morphy to Fischer: A history of the World Chess Championship. B T Batsford. 1973. The World Chess Championship: A History. Macmillan. 1973. General series *Batsford Chess Books **Discovering Chess Series. B T Batsford. Periodicals See [[Universal Bibliography/Periodicals#Chess|Periodicals, Chess]] *British Chess Magazine Othello *Goro Hasegawa with Maxine Brady. How to Win at Othello. (A Harvest/HBJ book). Jove Publications. 1977. ISBN 0-15-642215-8. [https://books.google.co.uk/books?id=mTdhlwVeyCcC] Wargames *Battleground. Tyne Tees. (ITV). 1978. [Television]. 6 episodes, with Edward Woodward. **Laurie Taylor. "Attila the Hun invades Tyne Tees". TV Times. 1978. pp 28 & 29. **Terry Wise. "Battleground". Battle for Wargamers. June 1978. pp 261 & 262. *[[w:Game of War|Game of War]]. Channel 4. 1997. [Television]. ==Toys== Periodicals *Games & Toys: The Leading Trade Journal for Home & Export. (H Richard Simmons Limited). [https://books.google.co.uk/books?id=pMmbZ_JTnXYC] Google: editions:UO8GID_4Ck0C *Toys and Novelties. (Sporting Goods Pub Co). [https://books.google.co.uk/books?id=zKdAAQAAMAAJ] [https://archive.org/details/toys-and-novelties-volume-9-1913/page/n53/mode/1up] (Toys and Novelties Publishing Company) [https://archive.org/details/toys-and-novelties-volume-19-issue-no.-1-6-january-june-1922/page/n173/mode/1up]. Cf. "Harcourt To Buy Journals From Haire Publishing Co" [https://books.google.co.uk/books?id=J7hEAQAAIAAJ 194] Publishers Weekly 29 *[[w:Playthings (magazine)|Playthings: The National Magazine of the Toy Trade]]. (McCready Publishing Co) [https://books.google.co.uk/books?id=eCQHzuYWDY4C] [https://books.google.co.uk/books?id=mnRO5WFXDfEC]. (Geyer-McAllister Publications). Cf. "Playthings bought by Geyer-McAllister" [https://books.google.co.uk/books?id=itAaAQAAMAAJ 52] Industrial Marketing Capsule toys Gachapon *Tim Hornyak. [https://www.japantimes.co.jp/life/2017/08/19/lifestyle/gachapon-tracing-evolution-japans-colorful-toy-capsules/ Gachapon: Tracing the evolution of Japan’s colorful toy capsules]. The Japan Times. 19 August 2017. ==Cricket== See [[w:Bibliography of cricket]] *Peter Arnold and Peter Wynne-Thomas. The Complete Encyclopedia of Cricket. 2006. 4th Ed: 2011: [https://books.google.co.uk/books?id=2R_pXwAACAAJ]. **Peter Arnold. The Illustrated Encyclopedia of World Cricket. *Morgan. The Encyclopedia of World Cricket. 2007. [https://books.google.co.uk/books?id=gFCbkgEACAAJ] Scores and biographies *Marylebone Club Cricket Scores and Biographies. [https://books.google.co.uk/books?id=dl8IAAAAQAAJ&pg=PR3#v=onepage&q&f=false] **See [[w:Arthur Haygarth]] and [[w:Fred Lillywhite]] Periodicals *[[w:Cricket: A Weekly Record of the Game|Cricket: A Weekly Record of the Game]]. [https://books.google.co.uk/books?id=eX9QAAAAYAAJ&pg=PP7#v=onepage&q&f=false]. Australia *Malcolm Andrews. The Encyclopaedia of Australian Cricket. 1980. [https://catalogue.nla.gov.au/Record/1531463] *The Oxford Companion to Australian Cricket India *The Encyclopaedia of Indian Cricket, 1965. [https://books.google.com/books?id=CE4Joad6iwAC] [Includes biographies] Annuals *[[w:Indian Cricket (annual)|Indian Cricket]]. [https://books.google.co.uk/books?id=ioRLAAAAYAAJ 1966]. ===Cricketers=== Cricketers, including biographical dictionaries and collections of biographies *[[w:ESPNcricinfo|ESPNcricinfo]] *[[w:CricketArchive|CricketArchive]] *John Arlott's Book of Cricketers. 1979. [https://books.google.co.uk/books?id=8-WBAAAAMAAJ] *World Cricketers: A Biographical Dictionary [https://books.google.com/books?id=IpBLAAAAYAAJ] *Carr's Dictionary of Extraordinary Cricketers. 1977. Aurum Press. 2005. [https://books.google.com/books?id=CfwsAAAACAAJ] *Sproat. Debrett's Cricketers' Who's Who. 1980. *S Canynge Caple. The Cricketer's Who's Who. Williams. Lincoln. 1934. *Cricket Who's Who: The Cricket Blue Book. 1909. [https://catalogue.nla.gov.au/Record/119715]. 1912. Bibliography: [https://books.google.co.uk/books?id=IjQyAQAAMAAJ] *Who's Who in Test Cricket: A Biographical Dictionary of Test Cricketers [https://books.google.com/books?id=5uF5PQAACAAJ] *Frindall. England Test Cricketers: The Complete Record from 1877. 1989. [https://books.google.com/books?id=2zHYLIW7h9UC] *Brooke. The Collins Who's Who of English First-Class Cricket, 1945-1984. 1985. [https://books.google.com/books?id=NGSPAAAACAAJ]. Review: [https://books.google.co.uk/books?id=iHMsAAAAYAAJ]. Commentary: [https://books.google.co.uk/books?id=wPg5AQAAIAAJ] Gloucestershire *Gloucestershire Cricketers, 1870-1979. (ACS Cricketers Series [https://archive.acscricket.com/cricketers_series/index.html]). The Association of Cricket Statisticians. Cleethorpes. 1979. [https://archive.acscricket.com/cricketers_series/gloucestershire_cricketers_1870-1979/index.html] *Rex Pogson. Gloucestershire Cricket and Cricketers, 1919-1939. Lytham St Annes. 1944. Catalogues: [https://catalogue.nla.gov.au/Record/850643] [https://books.google.co.uk/books?id=CS83vXlB1ZIC] [https://www.worldcat.org/title/504354999]. Also printed as microfilm: [https://books.google.co.uk/books?id=iqXeDTKUEl4C]. *Dean Hayes. Gloucestershire Cricketing Greats: 46 of the Best Cricketers for Gloucestershire. Tunbridge Wells. 1990. Catalogues: [https://books.google.co.uk/books?id=OmsqAQAAIAAJ] [https://www.worldcat.org/title/25202795] Australia *The A-Z of Australian Cricketers [https://books.google.com/books?id=w-0zAAAACAAJ] *Piesse. Encyclopedia of Australian Cricket Players. 2012. [https://books.google.com/books?id=Jsh4MAEACAAJ] *C P Moody. Australian Cricket and Cricketers 1856-1893-4. Melbourne. 1894. *Jack Pollard. Australian Cricket: The Game and the Players. Hodder and Stoughton. ABC Books. Sydney. Lane Cove, New South Wales. 1982. Angus & Robertson. London. North Ryde, New South Wales. Sydney. Revised Ed: 1988. Commentary: [https://books.google.co.uk/books?id=WotYAAAAYAAJ]. Review: [https://books.google.co.uk/books?id=KzNYAAAAMAAJ]. ==Geology== *Read and Watson. Introduction to Geology. Macmillan Education. 1962. 2nd Ed: 1968. Volume 1: Principles. Volume 2: Earth History. ==Mineralogy== *Bibliography of Mineralogy for 1886. Annual Report of the Board of Regents of the Smithsonian Institution. Year Ending 30 June 1887. 1889. Pages [https://books.google.co.uk/books?id=wDcWAAAAYAAJ&pg=PA473#v=onepage&q&f=false 473] to 476. *Battey, Maurice Hugh. Mineralogy for students. Oliver & Boyd. 1972. 2nd Ed. Longman. 1981. ==Paper== See [[s:Category:Paper]] *Surface. Bibliography of the Pulp and Paper Industries. Forest Service. Bulletin 123. 1913. [https://archive.org/details/bibliographyofpu12surf] *West. Reading List on Papermaking Materials. 1920 to 1921. [https://archive.org/details/readinglistonpa00westgoog] [https://archive.org/details/readinglistonpa01westgoog] ==Books== *British Book News [https://books.google.co.uk/books?id=2oFTAAAAIAAJ] *Australasian Book News and Literary Journal. Australasian Book News and Library Journal. [https://books.google.co.uk/books?id=QVQPAQAAIAAJ] *Book News. 1882 to 1918. (John Wanamaker). Called "Book News Monthly" from 1906. [https://books.google.co.uk/books?id=KtwRAAAAYAAJ&pg=PP7#v=onepage&q&f=false] *Stechert-Hafner Book News [https://books.google.co.uk/books?id=BmDqAAAAMAAJ] *U.S.A. Book News [https://books.google.co.uk/books?id=36gVAQAAIAAJ] *Branch Library Book News. [https://books.google.co.uk/books?id=NM8aAAAAMAAJ] *Hungarian Book Review [https://books.google.co.uk/books?id=6U85AQAAIAAJ] *Soviet Book News. (Earl Browder). 1947 [https://books.google.co.uk/books?id=QrXQ6LYSOF4C] *Miniature Book News. [https://books.google.co.uk/books?id=MascAQAAMAAJ] Rare *Berger. Rare Books and Special Collections. American Library Association. 2014. [https://books.google.co.uk/books?id=IFUangEACAAJ] Printed *Annual Bibliography of the History of the Printed Book and Libraries. [https://books.google.co.uk/books?id=GLigoebhrd8C&pg=PP1#v=onepage&q&f=false vol 30] [https://books.google.co.uk/books?id=UBN-IUZlF4gC&pg=PP1#v=onepage&q&f=false vol 31] ==Paperback and Paperbound== *Swados, "Paper Books: What do they Promise?" (1953) [https://books.google.co.uk/books?id=TwaJtQzwj1gC 173] The Nation 114 *Wagman, "The Paperbound Book Business" (1957) 9 Michigan Business Review [https://books.google.co.uk/books?id=9pA8uolQjnkC&pg=RA4-PA9#v=onepage&q&f=false 9] (No 5, November) ==Science== *Lafferty and Rowe. The Hutchinson Dictionary of Science. Helicon Publishing. 1993. 2nd Ed: 1998. ==Entertainment== *The Directory (The Times, 1996 onwards) Commentary: [https://www.marketingweek.com/as-times-starts-listings-supplement/] ==Fiction== ===Characters=== *Bowker's Guide to Characters in Fiction [https://books.google.co.uk/books?id=rVjlAAAAMAAJ] *William Freeman. Dictionary of Fictional Characters. [https://books.google.co.uk/books?id=sRYuAAAAMAAJ] *Martin Seymour-Smith. Dictionary of Fictional Characters. The Writer, Inc. [https://books.google.co.uk/books?id=MoQYAAAAIAAJ] *Cyclopedia of Literary Characters. Salem Press. Rev Ed: 1998. [https://books.google.co.uk/books?id=IJMYAAAAIAAJ] *Larousse Dictionary of Literary Characters. [https://books.google.co.uk/books?id=hq4UAQAAIAAJ] *Chambers Dictionary of Literary Characters. [https://books.google.com/books?id=KWNMPwAACAAJ] *Michael D Sollars. Dictionary of Literary Characters. [https://books.google.co.uk/books?id=IJddAQAACAAJ] ==Comedy and tragedy== *E H Mikhail. Comedy and Tragedy: A Bibliography of Critical Studies. 1972. [https://books.google.co.uk/books?id=5bgEAAAAMAAJ] ===Comedy=== *James E Evans. Comedy: An Annotated Bibliography of Theory and Criticism. 1987. [https://books.google.co.uk/books?id=s9ZgthAs1t8C&pg=PP1#v=onepage&q&f=false] *Meghan Duffy (ed). Daniel Gerould (senior ed). Comedy: A Bibliography of Critical Studies in English on the Theory and Practice of Comedy in Drama, Theatre, and Performance. New York. 2006. [https://books.google.co.uk/books?id=fI0jAQAAIAAJ] *David Humphrey. The Time of Laughter: Comedy and the Media Cultures of Japan. University of Michigan Press. 2023. [https://books.google.co.uk/books?id=CkTLEAAAQBAJ&pg=PP1#v=onepage&q&f=false] ==Radio and television== *William E McCavitt. Radio and Television: A Selected Annotated Bibliography. Scarecrow Press. 1978. [https://books.google.com/books?id=DAQVAQAAIAAJ] *Gertrude G Broderick. Radio and Television Bibliography. Bulletin 1948 No 17. [https://books.google.co.uk/books?id=yIPP1dsmNtAC&pg=PP5#v=onepage&q&f=false] *Patricia Beall Hamill. Radio and Television: A Selected Bibliography. Bulletin 1960 No 25. [https://books.google.co.uk/books?id=AJhGAQAAMAAJ&pg=PP3#v=onepage&q&f=false] ==Film and television== *Jean Mitry. Bibliographie internationale du cinéma et de la télévision. 1966 onwards. [https://books.google.co.uk/books?id=SC3gAAAAMAAJ] Commentary: A Reference Guide for English Studies, [https://books.google.co.uk/books?id=h_wfYKnMfOkC&pg=PA439#v=onepage&q&f=false p 439]. ==Broadcasting== *Iwao Nakajima. The Broadcasting Industry in Japan: Its Historical, Legal and Economic Aspects. (International Studies of Broadcasting). NHK Radio and TV Culture Research Institute. 1971. [https://books.google.co.uk/books?id=uZ4vAQAAIAAJ] *50 Years of Japanese Broadcasting. NHK. 1977. [https://books.google.com/books?id=FoSKqY7tG-AC] *The History of Broadcasting in Japan. Nippon Hoso Kyokai. 1967. [https://books.google.com/books?id=I1IHAQAAIAAJ] *Hitoshi Mitomo and Mikio Kimura. Broadcasting in Japan: Challenges and Opportunities. [https://books.google.co.uk/books?id=7hCcEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Broadcasting in Japan: Case-studies on Broadcasting Systems. 1978. 2011. [https://books.google.co.uk/books?id=Ez5ZBwAAQBAJ&pg=PR4#v=onepage&q&f=false] *"World's First TV Stereo Broadcasts in Japan". Japan Report. (vol 25, no 3). 1 February 1979. p 4. *Broadcasting in Japan: A Historical Review [https://books.google.com/books?id=uDYlAAAAMAAJ] Annuals *[[w:ja:日本民間放送年鑑|日本民間放送年鑑]] (Japan Commercial Broadcasting Yearbook). コーケン出版. [https://books.google.co.uk/books?id=vh22AAAAIAAJ 1997]. ==Animation== *John Halas and Roger Manvell. The Technique of Film Animation. 4th Ed: 1976. Focal Press. ISBN 0240509005. *Clements and McCarthy. The Anime Encyclopedia. 3rd Rev Ed: [https://books.google.co.uk/books?id=E03KBgAAQBAJ&pg=PA1958#v=onepage&q&f=false]. *[https://www.bloomberg.com/news/articles/2026-10-07/anime-s-popularity-is-rising-among-teen-fans Anime is winning over a new generation of fans]. Bloomberg. 9 October 2026. [https://www.japantimes.co.jp/culture/2026/10/09/entertainment-news/anime-survey-watching/ The Japan Times]. ==Colours== *Eiseman and Recker. Pantone: The 20th Century in Color. [https://books.google.co.uk/books?id=j3H7nSVS3UMC&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.theguardian.com/books/2011/nov/13/pantone-20th-century-color-review][https://www.theatlantic.com/entertainment/archive/2011/11/pantone-100-years-of-color/249016/][https://eu.vvdailypress.com/story/lifestyle/health-fitness/2012/01/16/color-reel-20th-century-s/37119883007/] ==Bilateral== Britain and Japan *Pearse. Companion to Japanese Britain and Ireland. In Print. 1991. [https://books.google.co.uk/books?id=KtAxAAAAIAAJ] ==Prehistoric life== Prehistoric animals *[[w:Michael Benton|Michael Benton]]. Prehistoric Animals: An A-Z Guide. Kingfisher Books. 1989. Derrydale Books, New York. 1989. [Illustrations: Jim Channell and Kevin Maddison.] *Ellis Owen. Prehistoric Animals: The Extraordinary Story of Life before Man. Octopus Books Limited. London. 1975. [Sculptures: Arthur Hayward.] Review: [https://books.google.co.uk/books?id=II-B8R-8Ov8C 17] Wildlife 422. Commentary: [https://books.google.co.uk/books?id=aUbYAAAAQBAJ&pg=PA269#v=onepage&q&f=false] [https://books.google.co.uk/books?id=jFNBAAAAIBAJ&pg=PA5#v=onepage&q&f=false]. **Prehistorische dieren: de geschiedenis van het leven vóór de mens. Translated by JJ Hoedeman. In den Toren, Baarn. Westland, Schoten. 1977. Commentary: [https://books.google.co.uk/books?id=ToVMAQAAIAAJ] **Les Animaux préhistoriques: l'extraordinaire histoire de la vie avant l'homme. Dinosaurs *Michael Benton. Dinosaurs: An A-Z Guide. Kingfisher Books. 1988. Derrydale Books, New York. 1988. [Illustrations: Jim Channell and Kevin Maddison.] ==Regions and other places== *JD Pearson. Oriental and Asian Bibliography: An Introduction with Some Reference to Africa. Archon Books. [https://books.google.com/books?id=P18ZAAAAMAAJ] *Theodore Besterman. A World Bibliography of Oriental Bibliographies. Revised by JD Pearson. Rowman and Littlefield. [https://books.google.co.uk/books?id=dvnfAAAAMAAJ] ==Culture and society== *Victoria Lyon Bestor, Theodore C Bestor and Akiko Yamagata (eds). Routledge Handbook of Japanese Culture and Society. 2011. [https://books.google.co.uk/books?id=0cBYffHp5L4C&pg=PP1#v=onepage&q&f=false] *Josef Kreiner and Hans-Dieter Ölschleger (eds). Japanese Culture and Society: Models of Interpretation. 1996. [https://books.google.co.uk/books?id=6skwAQAAIAAJ] *Dolores P Martinez. Modern Japanese Culture and Society. 2007. [https://books.google.co.uk/books?id=l6kMAQAAMAAJ vol 1] *C Andrew Gerstle. 18th Century Japan: Culture and Society. 1989. [https://books.google.co.uk/books?id=ViglDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Periodicals *Review of Japanese Culture and Society [https://books.google.co.uk/books?id=W_o-AQAAIAAJ] ==Clothes== *John Marshall. Make Your Own Japanese Clothes: Patterns and Ideas for Modern Wear. Kodansha International. 1988. [https://books.google.co.uk/books?id=ljOuuy0OO0UC&pg=PP1#v=onepage&q&f=false] ==Fashion== *Toby Slade. Japanese Fashion: A Cultural History. 2009. [https://books.google.co.uk/books?id=AInI19E48QIC&pg=PP3#v=onepage&q&f=false] *Bonnie English. Japanese Fashion Designers: The Work and Influence of Issey Miyake, Yohji Yamamotom, and Rei Kawakubo. 2011. [https://books.google.co.uk/books?id=6UMfAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sadao Hibi. Japanese Detail Fashion. Chronicle Books. 1989. [https://books.google.com/books?id=G-kIAQAAMAAJ] **Nihon No Dento Iro To Katachi. Graphic-sha Publishing. 1987. *Clémence Leleu. Japanese Fashion: A Journey from Tradition to Iconic Designers, Street Style to Global Influence. Translated by Andrea Reece. 2026. [https://books.google.co.uk/books?id=j2_MEQAAQBAJ&pg=PA1#v=onepage&q&f=false] **Mode Japonaise. 2025. Kimono *Sheila Cliffe. The Social Life of Kimono: Japanese Fashion Past and Present. 2017. [https://books.google.co.uk/books?id=7G6xDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Anna Jackson (ed). Kimono: The Art and Evolution of Japanese Fashion. V&A Publishing. 2010. [https://books.google.co.uk/books?id=8gtMAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/kimono-the-art-and-evolution-of-japanese-fashion-the-khalili-collections/oclc/917375803]. ==Documents== *Pages from History: Ten Documents that changed the World AD 1000 to AD 2000. The Sunday Times. (10 pull out supplements and folder). *Scott Christianson. 100 Documents That Changed the World: From Magna Carta to WikiLeaks. [https://books.google.co.uk/books?id=rxiUEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Joseph Janes. Documents That Changed the Way We Live. 2017. [https://books.google.co.uk/books?id=a6-zDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Graham Stewart. Britannia: 100 Documents that Shaped a Nation. [https://books.google.co.uk/books?id=XUuL3xyZ9k8C&pg=PA1#v=onepage&q&f=false] ==See also== *[[Bibliography]] ==Notes== {{Reflist}} {{subpagesif}} [[Category:Bibliographies]] [[Category:Research]] 5fxwk7i9g6ig51slbv5v1ydzn7dpwdc 1 171302 2844119 2842770 2026-11-03T06:41:25Z James500 297601 /* */ Add 2844119 wikitext text/x-wiki Terms that sometimes appear in the titles of bibliographies, guides to literature/reference, etc: Bibliography, catalogue, guide, list, index, in print, books, literature, sources, works, material, titles, reference. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 13:54, 14 November 2020 (UTC) For future use: *Non-fiction, art, fine arts, music, painting, sculpture, government, politics, philosophy, theology, religion, agriculture, forestry, fisheries, mining, manufacturing, technology, computer science, engineering, astronomy, physics, chemistry, biology, botany, natural history, prehistory, classics, anthropology, library science, librarianship, information studies, places, exploration, transport, Earth science, paleontology, official publications, universities, museums *Reprography, lithography, engraving *Bookbinding, early printed books, incunabula, manuscripts, papyri, papyrology *Languages, philology, ancient languages, medieval languages, dead languages, extinct languages *Writing, scripts, decipherment, alphabets, hieratic, demotic, hieroglyphic, figurative, ideographic, pictographic, cuneiform, inscriptions, stelae [stela], clay tablets, ostraka [ostrakon], papyrus, Minoan scripts, Linear A, Linear B *Indo-European languages; Italo-Celtic; Celtic languages *Latin, Greek, middle english, old english, anglo-norman [anglo-french], old french, old norse, old welsh, middle welsh, egyptian, hittite, luwian, lycian, akkadian, assyro-babylonian, elamite, amorite *Special libraries, national libraries, public libraries, university libraries, college libraries, school libraries, museum libraries *Archaeological theory, methodology, geology, stratigraphy etc. Field archaeology, behavioural archaeology *Tide tables *George Watson (ed). The Cambridge Bibliography of English Literature. Cambridge University Press. 1966. Volume 5. [https://books.google.co.uk/books?id=ebM6AAAAIAAJ] *[[s:Page:Cornwall (Mitton).djvu/230|"Some Books on Cornwall"]] *[[s:Author:Richard Colt Hoare]] *Yorkshire literature *Catalogues régionaux des incunables des bibliothèques publiques de France. [https://books.google.co.uk/books?id=K38VcZrjrj0C&pg=PA1#v=onepage&q&f=false] (regional catalogues of incunabula in the public libraries of France) * [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 16:24, 24 November 2019 (UTC) For future use: *Palmer's Index to The Times **editions:2Faf22c-dSwC *Routledge Handbook of Soviet and Russian Military Studies France: *Les Dictionnaires départementaux.  [https://books.google.co.uk/books?id=FZEwAQAAMAAJ] (series includes departmental biographical dictionaries) Creuse: *Guide des archives de la Creuse. 1972 [https://books.google.co.uk/books?id=eEsCAAAAMAAJ] *Joanne. Géographie du département de la Creuse. 1895 [https://books.google.co.uk/books?id=izeVfoPbIc8C] *Jamain. Le département de la Creuse: ses origines et sa pérennité [https://books.google.co.uk/books?id=kY-KMwcsJMEC&pg=PP1#v=onepage&q&f=false] *Annuaire du Département de la Creuse [https://books.google.co.uk/books?id=7wpBAAAAcAAJ&pg=PA1#v=onepage&q&f=false] *Journal du département de la Creuse (1807-1819) Bibliography: *[[w:Fredson Bowers|Bowers, Fredson]]. Principles of Bibliographical Description. Princeton University Press. 1949. Russell & Russell, New York. 1962. St Paul's Bibliographies (No 15). 1986. [https://books.google.co.uk/books?id=hA7hAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA525#v=onepage&q&f=false] *Bowers, Fredson. Bibliography and Textual Criticism. Clarendon Press, Oxford. 1964. [https://books.google.co.uk/books?id=zOFhNcQsihAC] Series: *Studies in Bibliography. Formerly "Papers in Bibliography". (Bibliographical Society of the University of Virginia). (1948 onwards) [https://books.google.co.uk/books?id=HGEUAAAAIAAJ] Commentary: "A History of Studies in Bibliography: The First Fifty Volumes" in The First Fifty Years, p 125 [https://books.google.co.uk/books?id=ZukVAQAAIAAJ] **Tanselle. Selected Studies in Bibliography. (Bibliographical Society of the University of Virginia) [https://books.google.co.uk/books?id=OnktAQAAIAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA526#v=onepage&q&f=false] Library associations: *IFLA. Functional Requirements for Bibliographic Records: Final Report. Saur. 1998. [https://books.google.co.uk/books?id=YW8hAAAAQBAJ&pg=PP4#v=onepage&q&f=false] *Patrick Le Boeuf (ed). Functional Requirements for Bibliographic Records (FRBR): Hype or Cure-All? Haworth Press. 2005. Routledge. 2013 [https://books.google.co.uk/books?id=Ka7hAQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Guidelines for Bibliographic Description of Reproductions. American Library Association. 1995. [https://books.google.co.uk/books?id=9zG6MGudKv8C&lpg=PP1&pg=PP1#v=onepage&q&f=false] Serials: *Serials Review Nostgia *Nostalgia cycles: [https://slate.com/culture/2012/04/the-golden-forty-year-rule-and-other-nostalgia-cycles-could-trends-possibly-return-every-40-years-20-years-and-12-15-years.html] [https://www.vice.com/en/article/how-the-20-year-trend-cycle-collapsed/] [https://www.npr.org/2022/03/01/1081115609/from-tumblrcore-to-2014core-the-nostalgia-loop-is-getting-smaller-and-faster] [https://www.theguardian.com/music/2023/apr/20/lets-go-round-again-the-ridiculous-rise-of-fifth-anniversary-vinyl-reissues] [https://www.independent.co.uk/life-style/fashion/gen-z-fashion-trends-tumblr-aesthetic-b2042247.html] *Stuck in the Seventies: 113 Things from the 1970s that Screwed Up the Twentysomething Generation. 1991. 1995. [https://books.google.co.uk/books?id=bZ03g5ylsaEC]. Review: [https://books.google.co.uk/books?id=6V4cAQAAMAAJ]. *1969: [https://books.google.co.uk/books?id=yuwjAAAAIBAJ&pg=PA25#v=onepage&q&f=false]. *Decade nostalgia [https://www.thetimes.com/uk/politics/article/rolling-back-the-years-qg8fs6sh06d] *[https://books.google.co.uk/books?id=COhVAAAAIBAJ&pg=PA20#v=onepage&q&f=false] [https://books.google.co.uk/books?id=Xf4rAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=nFtWAAAAIBAJ&pg=PA31#v=onepage&q&f=false] [https://books.google.co.uk/books?id=HqkfAAAAIBAJ&pg=PA27&article_id=4890,1282350#v=onepage&q&f=false] [https://books.google.co.uk/books?id=_WZGAAAAIBAJ&pg=PA29#v=onepage&q&f=false][https://books.google.co.uk/books?id=bzpHAAAAIBAJ&pg=PA27#v=onepage&q&f=false][https://books.google.co.uk/books?id=s1MgAAAAIBAJ&pg=PA7#v=onepage&q&f=false] *1970s: [https://books.google.co.uk/books?id=YT5QAAAAIBAJ&pg=PA8#v=onepage&q&f=false] *1990s Tamagotchi [https://www.nssmag.com/en/lifestyle/38204/tamagotchi-y2k-nostalgia] *[https://www.bbc.co.uk/programmes/m002g6k5 90s Nostalgia]. BBC. *[https://www.rd.com/article/tv-shows-like-stranger-things/ Stranger Things Is Over—Now What?! Here Are 7 Shows to Watch to Fill the Void]. Reader's Digest. 1 January 2026. Other *[[w:Haruko Ichikawa|Haruko Ichikawa]] (市河晴子 or 市川晴子) (1896-1943) **Japanese Lady in Europe. 1937. [https://books.google.co.uk/books?id=xi42AAAAMAAJ] (Japanese: 欧米の隅々) **Japanese Lady in America. First published 1938. [https://books.google.com/books?id=KQY7AQAAIAAJ] (Japanese: 米国の旅日本の旅) Search terms: *"press in japan" *"japanese press" *"press of japan" *"journalism of japan" *"journalism in japan" *"journalists in japan" *"japanese journalism" Generations *[https://www.vox.com/2018/8/15/17686668/millennials-explained Stop calling teenagers millennials]. Vox. 15 August 2018. Planet Earth *[[w:Encyclopedia of Earth|Encyclopedia of Earth]] Crime *The Concise Encyclopedia of Crime and Criminals [https://books.google.co.uk/books?id=uGTQAAAAIAAJ] Showa era *懐かしの昭和・平成流行事典: 2001-1945. [[w:ja:新人物往来社|Shin-Jinbutsuoraisha]]. 2002. [https://books.google.co.uk/books?id=QCYyAQAAIAAJ] *少年ブーム～昭和レトロの流行もの. 串間努著. 2003. [https://books.google.co.uk/books?id=sK0nAQAAIAAJ] Fading of the Showa era *[https://www.theguardian.com/world/gallery/2025/dec/02/fading-japan-showa-era-in-pictures The fading of Japan’s Shōwa era – in pictures]. The Guardian. 2 December 2025. *[https://soranews24.com/2026/04/02/fading-tokyo-searching-for-signs-of-the-showa-era-as-local-neighborhoods-evolvephotos/ Fading Tokyo – Searching for signs of the Showa era as local neighborhoods evolve]. SoraNews24. 2 April 2026. Showa retro packaging *[[w:ja:初見健一|Kenichi Hatsumi]]. 昭和のレトロパッケージ. [[w:ja:グラフィック社|Graphic-sha]]. 2016. ISBN 9784766128741. [https://books.google.co.uk/books?id=Ch5wjwEACAAJ] Shakespeare *Demmon (compiler). Catalogue of the Shakespearian Books and Pamphlets in the Joseph Crosby Library. 1885. [https://books.google.co.uk/books?id=GSZLAQAAMAAJ&pg=PA1#v=onepage&q&f=false]. *Shakespeare the Man and the Book. Part 2: Occasional Papers. 1881. [https://books.google.co.uk/books?id=gfUUAAAAYAAJ&pg=PP11#v=onepage&q&f=false] *Best films: [https://www.empireonline.com/movies/features/best-movies-of-all-time-us/] Survivability of past atmosphere *[https://news.climate.columbia.edu/2022/09/20/you-asked-dinosaurs-survived-when-co2-was-extremely-high-why-cant-humans/ You Asked: Dinosaurs Survived When CO2 Was Extremely High. Why Can’t Humans?] Columbia Climate School. 20 September 2022. *[https://www.sciencefocus.com/planet-earth/how-far-back-in-time-would-we-be-able-to-go-and-still-breathe-our-planets-air How far back in time would we be able to go and still breathe our planet's air?] BBC Science Focus. *[https://gizmodo.com/the-one-basic-fact-about-history-that-time-travelers-al-1652649190 The One Basic Fact About History That Time Travelers Always Forget]. io9. Gizmodo. 31 October 2014. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 10:55, 26 October 2020 (UTC) d98l3zs9ast2bnua44u6wzg7oq0pldt 2844121 2844119 2026-11-03T06:50:05Z James500 297601 /* */ Cut 2844121 wikitext text/x-wiki Terms that sometimes appear in the titles of bibliographies, guides to literature/reference, etc: Bibliography, catalogue, guide, list, index, in print, books, literature, sources, works, material, titles, reference. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 13:54, 14 November 2020 (UTC) For future use: *Non-fiction, art, fine arts, music, painting, sculpture, government, politics, philosophy, theology, religion, agriculture, forestry, fisheries, mining, manufacturing, technology, computer science, engineering, astronomy, physics, chemistry, biology, botany, natural history, prehistory, classics, anthropology, library science, librarianship, information studies, places, exploration, transport, Earth science, paleontology, official publications, universities, museums *Reprography, lithography, engraving *Bookbinding, early printed books, incunabula, manuscripts, papyri, papyrology *Languages, philology, ancient languages, medieval languages, dead languages, extinct languages *Writing, scripts, decipherment, alphabets, hieratic, demotic, hieroglyphic, figurative, ideographic, pictographic, cuneiform, inscriptions, stelae [stela], clay tablets, ostraka [ostrakon], papyrus, Minoan scripts, Linear A, Linear B *Indo-European languages; Italo-Celtic; Celtic languages *Latin, Greek, middle english, old english, anglo-norman [anglo-french], old french, old norse, old welsh, middle welsh, egyptian, hittite, luwian, lycian, akkadian, assyro-babylonian, elamite, amorite *Special libraries, national libraries, public libraries, university libraries, college libraries, school libraries, museum libraries *Archaeological theory, methodology, geology, stratigraphy etc. Field archaeology, behavioural archaeology *Tide tables *George Watson (ed). The Cambridge Bibliography of English Literature. Cambridge University Press. 1966. Volume 5. [https://books.google.co.uk/books?id=ebM6AAAAIAAJ] *[[s:Page:Cornwall (Mitton).djvu/230|"Some Books on Cornwall"]] *[[s:Author:Richard Colt Hoare]] *Yorkshire literature *Catalogues régionaux des incunables des bibliothèques publiques de France. [https://books.google.co.uk/books?id=K38VcZrjrj0C&pg=PA1#v=onepage&q&f=false] (regional catalogues of incunabula in the public libraries of France) * [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 16:24, 24 November 2019 (UTC) For future use: *Palmer's Index to The Times **editions:2Faf22c-dSwC *Routledge Handbook of Soviet and Russian Military Studies France: *Les Dictionnaires départementaux.  [https://books.google.co.uk/books?id=FZEwAQAAMAAJ] (series includes departmental biographical dictionaries) Creuse: *Guide des archives de la Creuse. 1972 [https://books.google.co.uk/books?id=eEsCAAAAMAAJ] *Joanne. Géographie du département de la Creuse. 1895 [https://books.google.co.uk/books?id=izeVfoPbIc8C] *Jamain. Le département de la Creuse: ses origines et sa pérennité [https://books.google.co.uk/books?id=kY-KMwcsJMEC&pg=PP1#v=onepage&q&f=false] *Annuaire du Département de la Creuse [https://books.google.co.uk/books?id=7wpBAAAAcAAJ&pg=PA1#v=onepage&q&f=false] *Journal du département de la Creuse (1807-1819) Bibliography: *[[w:Fredson Bowers|Bowers, Fredson]]. Principles of Bibliographical Description. Princeton University Press. 1949. Russell & Russell, New York. 1962. St Paul's Bibliographies (No 15). 1986. [https://books.google.co.uk/books?id=hA7hAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA525#v=onepage&q&f=false] *Bowers, Fredson. Bibliography and Textual Criticism. Clarendon Press, Oxford. 1964. [https://books.google.co.uk/books?id=zOFhNcQsihAC] Series: *Studies in Bibliography. Formerly "Papers in Bibliography". (Bibliographical Society of the University of Virginia). (1948 onwards) [https://books.google.co.uk/books?id=HGEUAAAAIAAJ] Commentary: "A History of Studies in Bibliography: The First Fifty Volumes" in The First Fifty Years, p 125 [https://books.google.co.uk/books?id=ZukVAQAAIAAJ] **Tanselle. Selected Studies in Bibliography. (Bibliographical Society of the University of Virginia) [https://books.google.co.uk/books?id=OnktAQAAIAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA526#v=onepage&q&f=false] Library associations: *IFLA. Functional Requirements for Bibliographic Records: Final Report. Saur. 1998. [https://books.google.co.uk/books?id=YW8hAAAAQBAJ&pg=PP4#v=onepage&q&f=false] *Patrick Le Boeuf (ed). Functional Requirements for Bibliographic Records (FRBR): Hype or Cure-All? Haworth Press. 2005. Routledge. 2013 [https://books.google.co.uk/books?id=Ka7hAQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Guidelines for Bibliographic Description of Reproductions. American Library Association. 1995. [https://books.google.co.uk/books?id=9zG6MGudKv8C&lpg=PP1&pg=PP1#v=onepage&q&f=false] Serials: *Serials Review Nostgia *Nostalgia cycles: [https://slate.com/culture/2012/04/the-golden-forty-year-rule-and-other-nostalgia-cycles-could-trends-possibly-return-every-40-years-20-years-and-12-15-years.html] [https://www.vice.com/en/article/how-the-20-year-trend-cycle-collapsed/] [https://www.npr.org/2022/03/01/1081115609/from-tumblrcore-to-2014core-the-nostalgia-loop-is-getting-smaller-and-faster] [https://www.theguardian.com/music/2023/apr/20/lets-go-round-again-the-ridiculous-rise-of-fifth-anniversary-vinyl-reissues] [https://www.independent.co.uk/life-style/fashion/gen-z-fashion-trends-tumblr-aesthetic-b2042247.html] *Stuck in the Seventies: 113 Things from the 1970s that Screwed Up the Twentysomething Generation. 1991. 1995. [https://books.google.co.uk/books?id=bZ03g5ylsaEC]. Review: [https://books.google.co.uk/books?id=6V4cAQAAMAAJ]. *1969: [https://books.google.co.uk/books?id=yuwjAAAAIBAJ&pg=PA25#v=onepage&q&f=false]. *[https://books.google.co.uk/books?id=COhVAAAAIBAJ&pg=PA20#v=onepage&q&f=false] [https://books.google.co.uk/books?id=Xf4rAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=nFtWAAAAIBAJ&pg=PA31#v=onepage&q&f=false] [https://books.google.co.uk/books?id=HqkfAAAAIBAJ&pg=PA27&article_id=4890,1282350#v=onepage&q&f=false] [https://books.google.co.uk/books?id=_WZGAAAAIBAJ&pg=PA29#v=onepage&q&f=false][https://books.google.co.uk/books?id=bzpHAAAAIBAJ&pg=PA27#v=onepage&q&f=false][https://books.google.co.uk/books?id=s1MgAAAAIBAJ&pg=PA7#v=onepage&q&f=false] *1970s: [https://books.google.co.uk/books?id=YT5QAAAAIBAJ&pg=PA8#v=onepage&q&f=false] *1990s Tamagotchi [https://www.nssmag.com/en/lifestyle/38204/tamagotchi-y2k-nostalgia] *[https://www.bbc.co.uk/programmes/m002g6k5 90s Nostalgia]. BBC. *[https://www.rd.com/article/tv-shows-like-stranger-things/ Stranger Things Is Over—Now What?! Here Are 7 Shows to Watch to Fill the Void]. Reader's Digest. 1 January 2026. Other *[[w:Haruko Ichikawa|Haruko Ichikawa]] (市河晴子 or 市川晴子) (1896-1943) **Japanese Lady in Europe. 1937. [https://books.google.co.uk/books?id=xi42AAAAMAAJ] (Japanese: 欧米の隅々) **Japanese Lady in America. First published 1938. [https://books.google.com/books?id=KQY7AQAAIAAJ] (Japanese: 米国の旅日本の旅) Search terms: *"press in japan" *"japanese press" *"press of japan" *"journalism of japan" *"journalism in japan" *"journalists in japan" *"japanese journalism" Generations *[https://www.vox.com/2018/8/15/17686668/millennials-explained Stop calling teenagers millennials]. Vox. 15 August 2018. Planet Earth *[[w:Encyclopedia of Earth|Encyclopedia of Earth]] Crime *The Concise Encyclopedia of Crime and Criminals [https://books.google.co.uk/books?id=uGTQAAAAIAAJ] Showa era *懐かしの昭和・平成流行事典: 2001-1945. [[w:ja:新人物往来社|Shin-Jinbutsuoraisha]]. 2002. [https://books.google.co.uk/books?id=QCYyAQAAIAAJ] *少年ブーム～昭和レトロの流行もの. 串間努著. 2003. [https://books.google.co.uk/books?id=sK0nAQAAIAAJ] Fading of the Showa era *[https://www.theguardian.com/world/gallery/2025/dec/02/fading-japan-showa-era-in-pictures The fading of Japan’s Shōwa era – in pictures]. The Guardian. 2 December 2025. *[https://soranews24.com/2026/04/02/fading-tokyo-searching-for-signs-of-the-showa-era-as-local-neighborhoods-evolvephotos/ Fading Tokyo – Searching for signs of the Showa era as local neighborhoods evolve]. SoraNews24. 2 April 2026. Showa retro packaging *[[w:ja:初見健一|Kenichi Hatsumi]]. 昭和のレトロパッケージ. [[w:ja:グラフィック社|Graphic-sha]]. 2016. ISBN 9784766128741. [https://books.google.co.uk/books?id=Ch5wjwEACAAJ] Shakespeare *Demmon (compiler). Catalogue of the Shakespearian Books and Pamphlets in the Joseph Crosby Library. 1885. [https://books.google.co.uk/books?id=GSZLAQAAMAAJ&pg=PA1#v=onepage&q&f=false]. *Shakespeare the Man and the Book. Part 2: Occasional Papers. 1881. [https://books.google.co.uk/books?id=gfUUAAAAYAAJ&pg=PP11#v=onepage&q&f=false] *Best films: [https://www.empireonline.com/movies/features/best-movies-of-all-time-us/] Survivability of past atmosphere *[https://news.climate.columbia.edu/2022/09/20/you-asked-dinosaurs-survived-when-co2-was-extremely-high-why-cant-humans/ You Asked: Dinosaurs Survived When CO2 Was Extremely High. Why Can’t Humans?] Columbia Climate School. 20 September 2022. *[https://www.sciencefocus.com/planet-earth/how-far-back-in-time-would-we-be-able-to-go-and-still-breathe-our-planets-air How far back in time would we be able to go and still breathe our planet's air?] BBC Science Focus. *[https://gizmodo.com/the-one-basic-fact-about-history-that-time-travelers-al-1652649190 The One Basic Fact About History That Time Travelers Always Forget]. io9. Gizmodo. 31 October 2014. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 10:55, 26 October 2020 (UTC) i4ysfd12zqtk9f4p6esm6khb86h6cjo 2844128 2844121 2026-11-03T07:47:31Z James500 297601 /* */ Add 2844128 wikitext text/x-wiki Terms that sometimes appear in the titles of bibliographies, guides to literature/reference, etc: Bibliography, catalogue, guide, list, index, in print, books, literature, sources, works, material, titles, reference. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 13:54, 14 November 2020 (UTC) For future use: *Non-fiction, art, fine arts, music, painting, sculpture, government, politics, philosophy, theology, religion, agriculture, forestry, fisheries, mining, manufacturing, technology, computer science, engineering, astronomy, physics, chemistry, biology, botany, natural history, prehistory, classics, anthropology, library science, librarianship, information studies, places, exploration, transport, Earth science, paleontology, official publications, universities, museums *Reprography, lithography, engraving *Bookbinding, early printed books, incunabula, manuscripts, papyri, papyrology *Languages, philology, ancient languages, medieval languages, dead languages, extinct languages *Writing, scripts, decipherment, alphabets, hieratic, demotic, hieroglyphic, figurative, ideographic, pictographic, cuneiform, inscriptions, stelae [stela], clay tablets, ostraka [ostrakon], papyrus, Minoan scripts, Linear A, Linear B *Indo-European languages; Italo-Celtic; Celtic languages *Latin, Greek, middle english, old english, anglo-norman [anglo-french], old french, old norse, old welsh, middle welsh, egyptian, hittite, luwian, lycian, akkadian, assyro-babylonian, elamite, amorite *Special libraries, national libraries, public libraries, university libraries, college libraries, school libraries, museum libraries *Archaeological theory, methodology, geology, stratigraphy etc. Field archaeology, behavioural archaeology *Tide tables *George Watson (ed). The Cambridge Bibliography of English Literature. Cambridge University Press. 1966. Volume 5. [https://books.google.co.uk/books?id=ebM6AAAAIAAJ] *[[s:Page:Cornwall (Mitton).djvu/230|"Some Books on Cornwall"]] *[[s:Author:Richard Colt Hoare]] *Yorkshire literature *Catalogues régionaux des incunables des bibliothèques publiques de France. [https://books.google.co.uk/books?id=K38VcZrjrj0C&pg=PA1#v=onepage&q&f=false] (regional catalogues of incunabula in the public libraries of France) * [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 16:24, 24 November 2019 (UTC) For future use: *Palmer's Index to The Times **editions:2Faf22c-dSwC *Routledge Handbook of Soviet and Russian Military Studies France: *Les Dictionnaires départementaux.  [https://books.google.co.uk/books?id=FZEwAQAAMAAJ] (series includes departmental biographical dictionaries) Creuse: *Guide des archives de la Creuse. 1972 [https://books.google.co.uk/books?id=eEsCAAAAMAAJ] *Joanne. Géographie du département de la Creuse. 1895 [https://books.google.co.uk/books?id=izeVfoPbIc8C] *Jamain. Le département de la Creuse: ses origines et sa pérennité [https://books.google.co.uk/books?id=kY-KMwcsJMEC&pg=PP1#v=onepage&q&f=false] *Annuaire du Département de la Creuse [https://books.google.co.uk/books?id=7wpBAAAAcAAJ&pg=PA1#v=onepage&q&f=false] *Journal du département de la Creuse (1807-1819) Bibliography: *[[w:Fredson Bowers|Bowers, Fredson]]. Principles of Bibliographical Description. Princeton University Press. 1949. Russell & Russell, New York. 1962. St Paul's Bibliographies (No 15). 1986. [https://books.google.co.uk/books?id=hA7hAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA525#v=onepage&q&f=false] *Bowers, Fredson. Bibliography and Textual Criticism. Clarendon Press, Oxford. 1964. [https://books.google.co.uk/books?id=zOFhNcQsihAC] Series: *Studies in Bibliography. Formerly "Papers in Bibliography". (Bibliographical Society of the University of Virginia). (1948 onwards) [https://books.google.co.uk/books?id=HGEUAAAAIAAJ] Commentary: "A History of Studies in Bibliography: The First Fifty Volumes" in The First Fifty Years, p 125 [https://books.google.co.uk/books?id=ZukVAQAAIAAJ] **Tanselle. Selected Studies in Bibliography. (Bibliographical Society of the University of Virginia) [https://books.google.co.uk/books?id=OnktAQAAIAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA526#v=onepage&q&f=false] Library associations: *IFLA. Functional Requirements for Bibliographic Records: Final Report. Saur. 1998. [https://books.google.co.uk/books?id=YW8hAAAAQBAJ&pg=PP4#v=onepage&q&f=false] *Patrick Le Boeuf (ed). Functional Requirements for Bibliographic Records (FRBR): Hype or Cure-All? Haworth Press. 2005. Routledge. 2013 [https://books.google.co.uk/books?id=Ka7hAQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Guidelines for Bibliographic Description of Reproductions. American Library Association. 1995. [https://books.google.co.uk/books?id=9zG6MGudKv8C&lpg=PP1&pg=PP1#v=onepage&q&f=false] Serials: *Serials Review Nostgia *Nostalgia cycles: [https://slate.com/culture/2012/04/the-golden-forty-year-rule-and-other-nostalgia-cycles-could-trends-possibly-return-every-40-years-20-years-and-12-15-years.html] [https://www.vice.com/en/article/how-the-20-year-trend-cycle-collapsed/] [https://www.npr.org/2022/03/01/1081115609/from-tumblrcore-to-2014core-the-nostalgia-loop-is-getting-smaller-and-faster] [https://www.theguardian.com/music/2023/apr/20/lets-go-round-again-the-ridiculous-rise-of-fifth-anniversary-vinyl-reissues] [https://www.independent.co.uk/life-style/fashion/gen-z-fashion-trends-tumblr-aesthetic-b2042247.html] *Stuck in the Seventies: 113 Things from the 1970s that Screwed Up the Twentysomething Generation. 1991. 1995. [https://books.google.co.uk/books?id=bZ03g5ylsaEC]. Review: [https://books.google.co.uk/books?id=6V4cAQAAMAAJ]. *1969: [https://books.google.co.uk/books?id=yuwjAAAAIBAJ&pg=PA25#v=onepage&q&f=false]. *[https://books.google.co.uk/books?id=COhVAAAAIBAJ&pg=PA20#v=onepage&q&f=false] [https://books.google.co.uk/books?id=Xf4rAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=nFtWAAAAIBAJ&pg=PA31#v=onepage&q&f=false] [https://books.google.co.uk/books?id=HqkfAAAAIBAJ&pg=PA27&article_id=4890,1282350#v=onepage&q&f=false] [https://books.google.co.uk/books?id=_WZGAAAAIBAJ&pg=PA29#v=onepage&q&f=false][https://books.google.co.uk/books?id=bzpHAAAAIBAJ&pg=PA27#v=onepage&q&f=false][https://books.google.co.uk/books?id=s1MgAAAAIBAJ&pg=PA7#v=onepage&q&f=false] *1970s: [https://books.google.co.uk/books?id=YT5QAAAAIBAJ&pg=PA8#v=onepage&q&f=false] *1990s Tamagotchi [https://www.nssmag.com/en/lifestyle/38204/tamagotchi-y2k-nostalgia] *[https://www.bbc.co.uk/programmes/m002g6k5 90s Nostalgia]. BBC. *[https://www.rd.com/article/tv-shows-like-stranger-things/ Stranger Things Is Over—Now What?! Here Are 7 Shows to Watch to Fill the Void]. Reader's Digest. 1 January 2026. Other *[[w:Haruko Ichikawa|Haruko Ichikawa]] (市河晴子 or 市川晴子) (1896-1943) **Japanese Lady in Europe. 1937. [https://books.google.co.uk/books?id=xi42AAAAMAAJ] (Japanese: 欧米の隅々) **Japanese Lady in America. First published 1938. [https://books.google.com/books?id=KQY7AQAAIAAJ] (Japanese: 米国の旅日本の旅) Search terms: *"press in japan" *"japanese press" *"press of japan" *"journalism of japan" *"journalism in japan" *"journalists in japan" *"japanese journalism" Generations *[https://www.vox.com/2018/8/15/17686668/millennials-explained Stop calling teenagers millennials]. Vox. 15 August 2018. Planet Earth *[[w:Encyclopedia of Earth|Encyclopedia of Earth]] Crime *The Concise Encyclopedia of Crime and Criminals [https://books.google.co.uk/books?id=uGTQAAAAIAAJ] Showa era *懐かしの昭和・平成流行事典: 2001-1945. [[w:ja:新人物往来社|Shin-Jinbutsuoraisha]]. 2002. [https://books.google.co.uk/books?id=QCYyAQAAIAAJ] *少年ブーム～昭和レトロの流行もの. 串間努著. 2003. [https://books.google.co.uk/books?id=sK0nAQAAIAAJ] Fading of the Showa era *[https://www.theguardian.com/world/gallery/2025/dec/02/fading-japan-showa-era-in-pictures The fading of Japan’s Shōwa era – in pictures]. The Guardian. 2 December 2025. *[https://soranews24.com/2026/04/02/fading-tokyo-searching-for-signs-of-the-showa-era-as-local-neighborhoods-evolvephotos/ Fading Tokyo – Searching for signs of the Showa era as local neighborhoods evolve]. SoraNews24. 2 April 2026. Showa retro packaging *[[w:ja:初見健一|Kenichi Hatsumi]]. 昭和のレトロパッケージ. [[w:ja:グラフィック社|Graphic-sha]]. 2016. ISBN 9784766128741. [https://books.google.co.uk/books?id=Ch5wjwEACAAJ] Shotengai *[https://www.theguardian.com/world/2025/jul/18/cult-of-convenience-how-tokyos-retro-shotengai-arcades-are-falling-victim-to-gentrification ‘Cult of convenience’: how Tokyo’s retro shotengai arcades are falling victim to gentrification]. The Guardian. 18 July 2025. Shakespeare *Demmon (compiler). Catalogue of the Shakespearian Books and Pamphlets in the Joseph Crosby Library. 1885. [https://books.google.co.uk/books?id=GSZLAQAAMAAJ&pg=PA1#v=onepage&q&f=false]. *Shakespeare the Man and the Book. Part 2: Occasional Papers. 1881. [https://books.google.co.uk/books?id=gfUUAAAAYAAJ&pg=PP11#v=onepage&q&f=false] *Best films: [https://www.empireonline.com/movies/features/best-movies-of-all-time-us/] Survivability of past atmosphere *[https://news.climate.columbia.edu/2022/09/20/you-asked-dinosaurs-survived-when-co2-was-extremely-high-why-cant-humans/ You Asked: Dinosaurs Survived When CO2 Was Extremely High. Why Can’t Humans?] Columbia Climate School. 20 September 2022. *[https://www.sciencefocus.com/planet-earth/how-far-back-in-time-would-we-be-able-to-go-and-still-breathe-our-planets-air How far back in time would we be able to go and still breathe our planet's air?] BBC Science Focus. *[https://gizmodo.com/the-one-basic-fact-about-history-that-time-travelers-al-1652649190 The One Basic Fact About History That Time Travelers Always Forget]. io9. Gizmodo. 31 October 2014. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 10:55, 26 October 2020 (UTC) k7slbz3ckf4t0ddj63vqvd01hh7upwx 2844142 2844128 2026-11-03T08:58:28Z James500 297601 /* */ Add 2844142 wikitext text/x-wiki Terms that sometimes appear in the titles of bibliographies, guides to literature/reference, etc: Bibliography, catalogue, guide, list, index, in print, books, literature, sources, works, material, titles, reference. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 13:54, 14 November 2020 (UTC) For future use: *Non-fiction, art, fine arts, music, painting, sculpture, government, politics, philosophy, theology, religion, agriculture, forestry, fisheries, mining, manufacturing, technology, computer science, engineering, astronomy, physics, chemistry, biology, botany, natural history, prehistory, classics, anthropology, library science, librarianship, information studies, places, exploration, transport, Earth science, paleontology, official publications, universities, museums *Reprography, lithography, engraving *Bookbinding, early printed books, incunabula, manuscripts, papyri, papyrology *Languages, philology, ancient languages, medieval languages, dead languages, extinct languages *Writing, scripts, decipherment, alphabets, hieratic, demotic, hieroglyphic, figurative, ideographic, pictographic, cuneiform, inscriptions, stelae [stela], clay tablets, ostraka [ostrakon], papyrus, Minoan scripts, Linear A, Linear B *Indo-European languages; Italo-Celtic; Celtic languages *Latin, Greek, middle english, old english, anglo-norman [anglo-french], old french, old norse, old welsh, middle welsh, egyptian, hittite, luwian, lycian, akkadian, assyro-babylonian, elamite, amorite *Special libraries, national libraries, public libraries, university libraries, college libraries, school libraries, museum libraries *Archaeological theory, methodology, geology, stratigraphy etc. Field archaeology, behavioural archaeology *Tide tables *George Watson (ed). The Cambridge Bibliography of English Literature. Cambridge University Press. 1966. Volume 5. [https://books.google.co.uk/books?id=ebM6AAAAIAAJ] *[[s:Page:Cornwall (Mitton).djvu/230|"Some Books on Cornwall"]] *[[s:Author:Richard Colt Hoare]] *Yorkshire literature *Catalogues régionaux des incunables des bibliothèques publiques de France. [https://books.google.co.uk/books?id=K38VcZrjrj0C&pg=PA1#v=onepage&q&f=false] (regional catalogues of incunabula in the public libraries of France) * [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 16:24, 24 November 2019 (UTC) For future use: *Palmer's Index to The Times **editions:2Faf22c-dSwC *Routledge Handbook of Soviet and Russian Military Studies France: *Les Dictionnaires départementaux.  [https://books.google.co.uk/books?id=FZEwAQAAMAAJ] (series includes departmental biographical dictionaries) Creuse: *Guide des archives de la Creuse. 1972 [https://books.google.co.uk/books?id=eEsCAAAAMAAJ] *Joanne. Géographie du département de la Creuse. 1895 [https://books.google.co.uk/books?id=izeVfoPbIc8C] *Jamain. Le département de la Creuse: ses origines et sa pérennité [https://books.google.co.uk/books?id=kY-KMwcsJMEC&pg=PP1#v=onepage&q&f=false] *Annuaire du Département de la Creuse [https://books.google.co.uk/books?id=7wpBAAAAcAAJ&pg=PA1#v=onepage&q&f=false] *Journal du département de la Creuse (1807-1819) Bibliography: *[[w:Fredson Bowers|Bowers, Fredson]]. Principles of Bibliographical Description. Princeton University Press. 1949. Russell & Russell, New York. 1962. St Paul's Bibliographies (No 15). 1986. [https://books.google.co.uk/books?id=hA7hAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA525#v=onepage&q&f=false] *Bowers, Fredson. Bibliography and Textual Criticism. Clarendon Press, Oxford. 1964. [https://books.google.co.uk/books?id=zOFhNcQsihAC] Series: *Studies in Bibliography. Formerly "Papers in Bibliography". (Bibliographical Society of the University of Virginia). (1948 onwards) [https://books.google.co.uk/books?id=HGEUAAAAIAAJ] Commentary: "A History of Studies in Bibliography: The First Fifty Volumes" in The First Fifty Years, p 125 [https://books.google.co.uk/books?id=ZukVAQAAIAAJ] **Tanselle. Selected Studies in Bibliography. (Bibliographical Society of the University of Virginia) [https://books.google.co.uk/books?id=OnktAQAAIAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA526#v=onepage&q&f=false] Library associations: *IFLA. Functional Requirements for Bibliographic Records: Final Report. Saur. 1998. [https://books.google.co.uk/books?id=YW8hAAAAQBAJ&pg=PP4#v=onepage&q&f=false] *Patrick Le Boeuf (ed). Functional Requirements for Bibliographic Records (FRBR): Hype or Cure-All? Haworth Press. 2005. Routledge. 2013 [https://books.google.co.uk/books?id=Ka7hAQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Guidelines for Bibliographic Description of Reproductions. American Library Association. 1995. [https://books.google.co.uk/books?id=9zG6MGudKv8C&lpg=PP1&pg=PP1#v=onepage&q&f=false] Serials: *Serials Review Nostgia *Nostalgia cycles: [https://slate.com/culture/2012/04/the-golden-forty-year-rule-and-other-nostalgia-cycles-could-trends-possibly-return-every-40-years-20-years-and-12-15-years.html] [https://www.vice.com/en/article/how-the-20-year-trend-cycle-collapsed/] [https://www.npr.org/2022/03/01/1081115609/from-tumblrcore-to-2014core-the-nostalgia-loop-is-getting-smaller-and-faster] [https://www.theguardian.com/music/2023/apr/20/lets-go-round-again-the-ridiculous-rise-of-fifth-anniversary-vinyl-reissues] [https://www.independent.co.uk/life-style/fashion/gen-z-fashion-trends-tumblr-aesthetic-b2042247.html] *Stuck in the Seventies: 113 Things from the 1970s that Screwed Up the Twentysomething Generation. 1991. 1995. [https://books.google.co.uk/books?id=bZ03g5ylsaEC]. Review: [https://books.google.co.uk/books?id=6V4cAQAAMAAJ]. *1969: [https://books.google.co.uk/books?id=yuwjAAAAIBAJ&pg=PA25#v=onepage&q&f=false]. *[https://books.google.co.uk/books?id=COhVAAAAIBAJ&pg=PA20#v=onepage&q&f=false] [https://books.google.co.uk/books?id=Xf4rAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=nFtWAAAAIBAJ&pg=PA31#v=onepage&q&f=false] [https://books.google.co.uk/books?id=HqkfAAAAIBAJ&pg=PA27&article_id=4890,1282350#v=onepage&q&f=false] [https://books.google.co.uk/books?id=_WZGAAAAIBAJ&pg=PA29#v=onepage&q&f=false][https://books.google.co.uk/books?id=bzpHAAAAIBAJ&pg=PA27#v=onepage&q&f=false][https://books.google.co.uk/books?id=s1MgAAAAIBAJ&pg=PA7#v=onepage&q&f=false] *1970s: [https://books.google.co.uk/books?id=YT5QAAAAIBAJ&pg=PA8#v=onepage&q&f=false] *1990s Tamagotchi [https://www.nssmag.com/en/lifestyle/38204/tamagotchi-y2k-nostalgia] *[https://www.bbc.co.uk/programmes/m002g6k5 90s Nostalgia]. BBC. *[https://www.rd.com/article/tv-shows-like-stranger-things/ Stranger Things Is Over—Now What?! Here Are 7 Shows to Watch to Fill the Void]. Reader's Digest. 1 January 2026. *Anne Latreille. [https://trove.nla.gov.au/newspaper/article/47808279 "There's a Boom in Nostalgia"]. [[w:The Australian Women's Weekly|The Australian Women's Weekly]]. 29 March 1972. p 7. Other *[[w:Haruko Ichikawa|Haruko Ichikawa]] (市河晴子 or 市川晴子) (1896-1943) **Japanese Lady in Europe. 1937. [https://books.google.co.uk/books?id=xi42AAAAMAAJ] (Japanese: 欧米の隅々) **Japanese Lady in America. First published 1938. [https://books.google.com/books?id=KQY7AQAAIAAJ] (Japanese: 米国の旅日本の旅) Search terms: *"press in japan" *"japanese press" *"press of japan" *"journalism of japan" *"journalism in japan" *"journalists in japan" *"japanese journalism" Generations *[https://www.vox.com/2018/8/15/17686668/millennials-explained Stop calling teenagers millennials]. Vox. 15 August 2018. Planet Earth *[[w:Encyclopedia of Earth|Encyclopedia of Earth]] Crime *The Concise Encyclopedia of Crime and Criminals [https://books.google.co.uk/books?id=uGTQAAAAIAAJ] Showa era *懐かしの昭和・平成流行事典: 2001-1945. [[w:ja:新人物往来社|Shin-Jinbutsuoraisha]]. 2002. [https://books.google.co.uk/books?id=QCYyAQAAIAAJ] *少年ブーム～昭和レトロの流行もの. 串間努著. 2003. [https://books.google.co.uk/books?id=sK0nAQAAIAAJ] Fading of the Showa era *[https://www.theguardian.com/world/gallery/2025/dec/02/fading-japan-showa-era-in-pictures The fading of Japan’s Shōwa era – in pictures]. The Guardian. 2 December 2025. *[https://soranews24.com/2026/04/02/fading-tokyo-searching-for-signs-of-the-showa-era-as-local-neighborhoods-evolvephotos/ Fading Tokyo – Searching for signs of the Showa era as local neighborhoods evolve]. SoraNews24. 2 April 2026. Showa retro packaging *[[w:ja:初見健一|Kenichi Hatsumi]]. 昭和のレトロパッケージ. [[w:ja:グラフィック社|Graphic-sha]]. 2016. ISBN 9784766128741. [https://books.google.co.uk/books?id=Ch5wjwEACAAJ] Shotengai *[https://www.theguardian.com/world/2025/jul/18/cult-of-convenience-how-tokyos-retro-shotengai-arcades-are-falling-victim-to-gentrification ‘Cult of convenience’: how Tokyo’s retro shotengai arcades are falling victim to gentrification]. The Guardian. 18 July 2025. Shakespeare *Demmon (compiler). Catalogue of the Shakespearian Books and Pamphlets in the Joseph Crosby Library. 1885. [https://books.google.co.uk/books?id=GSZLAQAAMAAJ&pg=PA1#v=onepage&q&f=false]. *Shakespeare the Man and the Book. Part 2: Occasional Papers. 1881. [https://books.google.co.uk/books?id=gfUUAAAAYAAJ&pg=PP11#v=onepage&q&f=false] *Best films: [https://www.empireonline.com/movies/features/best-movies-of-all-time-us/] Survivability of past atmosphere *[https://news.climate.columbia.edu/2022/09/20/you-asked-dinosaurs-survived-when-co2-was-extremely-high-why-cant-humans/ You Asked: Dinosaurs Survived When CO2 Was Extremely High. Why Can’t Humans?] Columbia Climate School. 20 September 2022. *[https://www.sciencefocus.com/planet-earth/how-far-back-in-time-would-we-be-able-to-go-and-still-breathe-our-planets-air How far back in time would we be able to go and still breathe our planet's air?] BBC Science Focus. *[https://gizmodo.com/the-one-basic-fact-about-history-that-time-travelers-al-1652649190 The One Basic Fact About History That Time Travelers Always Forget]. io9. Gizmodo. 31 October 2014. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 10:55, 26 October 2020 (UTC) n1wrfmxqcoe2l9m9tu6esulzts12sze 2844144 2844142 2026-11-03T09:01:57Z James500 297601 /* */ Add 2844144 wikitext text/x-wiki Terms that sometimes appear in the titles of bibliographies, guides to literature/reference, etc: Bibliography, catalogue, guide, list, index, in print, books, literature, sources, works, material, titles, reference. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 13:54, 14 November 2020 (UTC) For future use: *Non-fiction, art, fine arts, music, painting, sculpture, government, politics, philosophy, theology, religion, agriculture, forestry, fisheries, mining, manufacturing, technology, computer science, engineering, astronomy, physics, chemistry, biology, botany, natural history, prehistory, classics, anthropology, library science, librarianship, information studies, places, exploration, transport, Earth science, paleontology, official publications, universities, museums *Reprography, lithography, engraving *Bookbinding, early printed books, incunabula, manuscripts, papyri, papyrology *Languages, philology, ancient languages, medieval languages, dead languages, extinct languages *Writing, scripts, decipherment, alphabets, hieratic, demotic, hieroglyphic, figurative, ideographic, pictographic, cuneiform, inscriptions, stelae [stela], clay tablets, ostraka [ostrakon], papyrus, Minoan scripts, Linear A, Linear B *Indo-European languages; Italo-Celtic; Celtic languages *Latin, Greek, middle english, old english, anglo-norman [anglo-french], old french, old norse, old welsh, middle welsh, egyptian, hittite, luwian, lycian, akkadian, assyro-babylonian, elamite, amorite *Special libraries, national libraries, public libraries, university libraries, college libraries, school libraries, museum libraries *Archaeological theory, methodology, geology, stratigraphy etc. Field archaeology, behavioural archaeology *Tide tables *George Watson (ed). The Cambridge Bibliography of English Literature. Cambridge University Press. 1966. Volume 5. [https://books.google.co.uk/books?id=ebM6AAAAIAAJ] *[[s:Page:Cornwall (Mitton).djvu/230|"Some Books on Cornwall"]] *[[s:Author:Richard Colt Hoare]] *Yorkshire literature *Catalogues régionaux des incunables des bibliothèques publiques de France. [https://books.google.co.uk/books?id=K38VcZrjrj0C&pg=PA1#v=onepage&q&f=false] (regional catalogues of incunabula in the public libraries of France) * [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 16:24, 24 November 2019 (UTC) For future use: *Palmer's Index to The Times **editions:2Faf22c-dSwC *Routledge Handbook of Soviet and Russian Military Studies France: *Les Dictionnaires départementaux.  [https://books.google.co.uk/books?id=FZEwAQAAMAAJ] (series includes departmental biographical dictionaries) Creuse: *Guide des archives de la Creuse. 1972 [https://books.google.co.uk/books?id=eEsCAAAAMAAJ] *Joanne. Géographie du département de la Creuse. 1895 [https://books.google.co.uk/books?id=izeVfoPbIc8C] *Jamain. Le département de la Creuse: ses origines et sa pérennité [https://books.google.co.uk/books?id=kY-KMwcsJMEC&pg=PP1#v=onepage&q&f=false] *Annuaire du Département de la Creuse [https://books.google.co.uk/books?id=7wpBAAAAcAAJ&pg=PA1#v=onepage&q&f=false] *Journal du département de la Creuse (1807-1819) Bibliography: *[[w:Fredson Bowers|Bowers, Fredson]]. Principles of Bibliographical Description. Princeton University Press. 1949. Russell & Russell, New York. 1962. St Paul's Bibliographies (No 15). 1986. [https://books.google.co.uk/books?id=hA7hAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA525#v=onepage&q&f=false] *Bowers, Fredson. Bibliography and Textual Criticism. Clarendon Press, Oxford. 1964. [https://books.google.co.uk/books?id=zOFhNcQsihAC] Series: *Studies in Bibliography. Formerly "Papers in Bibliography". (Bibliographical Society of the University of Virginia). (1948 onwards) [https://books.google.co.uk/books?id=HGEUAAAAIAAJ] Commentary: "A History of Studies in Bibliography: The First Fifty Volumes" in The First Fifty Years, p 125 [https://books.google.co.uk/books?id=ZukVAQAAIAAJ] **Tanselle. Selected Studies in Bibliography. (Bibliographical Society of the University of Virginia) [https://books.google.co.uk/books?id=OnktAQAAIAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA526#v=onepage&q&f=false] Library associations: *IFLA. Functional Requirements for Bibliographic Records: Final Report. Saur. 1998. [https://books.google.co.uk/books?id=YW8hAAAAQBAJ&pg=PP4#v=onepage&q&f=false] *Patrick Le Boeuf (ed). Functional Requirements for Bibliographic Records (FRBR): Hype or Cure-All? Haworth Press. 2005. Routledge. 2013 [https://books.google.co.uk/books?id=Ka7hAQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Guidelines for Bibliographic Description of Reproductions. American Library Association. 1995. [https://books.google.co.uk/books?id=9zG6MGudKv8C&lpg=PP1&pg=PP1#v=onepage&q&f=false] Serials: *Serials Review Nostgia *Nostalgia cycles: [https://slate.com/culture/2012/04/the-golden-forty-year-rule-and-other-nostalgia-cycles-could-trends-possibly-return-every-40-years-20-years-and-12-15-years.html] [https://www.vice.com/en/article/how-the-20-year-trend-cycle-collapsed/] [https://www.npr.org/2022/03/01/1081115609/from-tumblrcore-to-2014core-the-nostalgia-loop-is-getting-smaller-and-faster] [https://www.theguardian.com/music/2023/apr/20/lets-go-round-again-the-ridiculous-rise-of-fifth-anniversary-vinyl-reissues] [https://www.independent.co.uk/life-style/fashion/gen-z-fashion-trends-tumblr-aesthetic-b2042247.html] *Stuck in the Seventies: 113 Things from the 1970s that Screwed Up the Twentysomething Generation. 1991. 1995. [https://books.google.co.uk/books?id=bZ03g5ylsaEC]. Review: [https://books.google.co.uk/books?id=6V4cAQAAMAAJ]. *1969: [https://books.google.co.uk/books?id=yuwjAAAAIBAJ&pg=PA25#v=onepage&q&f=false]. *[https://books.google.co.uk/books?id=COhVAAAAIBAJ&pg=PA20#v=onepage&q&f=false] [https://books.google.co.uk/books?id=Xf4rAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=nFtWAAAAIBAJ&pg=PA31#v=onepage&q&f=false] [https://books.google.co.uk/books?id=HqkfAAAAIBAJ&pg=PA27&article_id=4890,1282350#v=onepage&q&f=false] [https://books.google.co.uk/books?id=_WZGAAAAIBAJ&pg=PA29#v=onepage&q&f=false][https://books.google.co.uk/books?id=bzpHAAAAIBAJ&pg=PA27#v=onepage&q&f=false][https://books.google.co.uk/books?id=s1MgAAAAIBAJ&pg=PA7#v=onepage&q&f=false] *1970s: [https://books.google.co.uk/books?id=YT5QAAAAIBAJ&pg=PA8#v=onepage&q&f=false] *1990s Tamagotchi [https://www.nssmag.com/en/lifestyle/38204/tamagotchi-y2k-nostalgia] *[https://www.bbc.co.uk/programmes/m002g6k5 90s Nostalgia]. BBC. *[https://www.rd.com/article/tv-shows-like-stranger-things/ Stranger Things Is Over—Now What?! Here Are 7 Shows to Watch to Fill the Void]. Reader's Digest. 1 January 2026. *Anne Latreille. [https://trove.nla.gov.au/newspaper/article/47808279 "There's a Boom in Nostalgia"]. [[w:The Australian Women's Weekly|The Australian Women's Weekly]]. 29 March 1972. p 7. *The Past in the Present: Proceedings of the 5th Annual British and Cultural Studies Conference, Oldenburg 1994 Other *[[w:Haruko Ichikawa|Haruko Ichikawa]] (市河晴子 or 市川晴子) (1896-1943) **Japanese Lady in Europe. 1937. [https://books.google.co.uk/books?id=xi42AAAAMAAJ] (Japanese: 欧米の隅々) **Japanese Lady in America. First published 1938. [https://books.google.com/books?id=KQY7AQAAIAAJ] (Japanese: 米国の旅日本の旅) Search terms: *"press in japan" *"japanese press" *"press of japan" *"journalism of japan" *"journalism in japan" *"journalists in japan" *"japanese journalism" Generations *[https://www.vox.com/2018/8/15/17686668/millennials-explained Stop calling teenagers millennials]. Vox. 15 August 2018. Planet Earth *[[w:Encyclopedia of Earth|Encyclopedia of Earth]] Crime *The Concise Encyclopedia of Crime and Criminals [https://books.google.co.uk/books?id=uGTQAAAAIAAJ] Showa era *懐かしの昭和・平成流行事典: 2001-1945. [[w:ja:新人物往来社|Shin-Jinbutsuoraisha]]. 2002. [https://books.google.co.uk/books?id=QCYyAQAAIAAJ] *少年ブーム～昭和レトロの流行もの. 串間努著. 2003. [https://books.google.co.uk/books?id=sK0nAQAAIAAJ] Fading of the Showa era *[https://www.theguardian.com/world/gallery/2025/dec/02/fading-japan-showa-era-in-pictures The fading of Japan’s Shōwa era – in pictures]. The Guardian. 2 December 2025. *[https://soranews24.com/2026/04/02/fading-tokyo-searching-for-signs-of-the-showa-era-as-local-neighborhoods-evolvephotos/ Fading Tokyo – Searching for signs of the Showa era as local neighborhoods evolve]. SoraNews24. 2 April 2026. Showa retro packaging *[[w:ja:初見健一|Kenichi Hatsumi]]. 昭和のレトロパッケージ. [[w:ja:グラフィック社|Graphic-sha]]. 2016. ISBN 9784766128741. [https://books.google.co.uk/books?id=Ch5wjwEACAAJ] Shotengai *[https://www.theguardian.com/world/2025/jul/18/cult-of-convenience-how-tokyos-retro-shotengai-arcades-are-falling-victim-to-gentrification ‘Cult of convenience’: how Tokyo’s retro shotengai arcades are falling victim to gentrification]. The Guardian. 18 July 2025. Shakespeare *Demmon (compiler). Catalogue of the Shakespearian Books and Pamphlets in the Joseph Crosby Library. 1885. [https://books.google.co.uk/books?id=GSZLAQAAMAAJ&pg=PA1#v=onepage&q&f=false]. *Shakespeare the Man and the Book. Part 2: Occasional Papers. 1881. [https://books.google.co.uk/books?id=gfUUAAAAYAAJ&pg=PP11#v=onepage&q&f=false] *Best films: [https://www.empireonline.com/movies/features/best-movies-of-all-time-us/] Survivability of past atmosphere *[https://news.climate.columbia.edu/2022/09/20/you-asked-dinosaurs-survived-when-co2-was-extremely-high-why-cant-humans/ You Asked: Dinosaurs Survived When CO2 Was Extremely High. Why Can’t Humans?] Columbia Climate School. 20 September 2022. *[https://www.sciencefocus.com/planet-earth/how-far-back-in-time-would-we-be-able-to-go-and-still-breathe-our-planets-air How far back in time would we be able to go and still breathe our planet's air?] BBC Science Focus. *[https://gizmodo.com/the-one-basic-fact-about-history-that-time-travelers-al-1652649190 The One Basic Fact About History That Time Travelers Always Forget]. io9. Gizmodo. 31 October 2014. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 10:55, 26 October 2020 (UTC) 8p55vl9pfe2b4aw1pvdqpn7atj85slc 0 203942 2843991 2843404 2026-11-02T16:46:18Z Young1lim 21186 /* Lambda Calculus */ 2843991 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20260915.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20261102.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] ge5s9723m342uf8vsedgev5j6quanst 0 214188 2844111 2614591 2026-11-03T06:02:44Z ~2026-58446-88 3113739 58 2844111 wikitext text/x-wiki 58 {| class="wikitable" |- ! Number !! Bengali !Transliteration |- | 0 || ০ শূন্য |Shnn no (Śūn'ya) |- | 1 || ১ এক |ak (Ēka) |- | 2 || ২ দুই |Dui (Du'i) |- | 3 || ৩ তিন |Tin (Tina) |- | 4 || ৪ চার |Char (Cāra) |- | 5 || ৫ পাঁচ |Paatch (Pām̐ca) |- | 6|| ৬ ছয় |Choy (Chaẏa) |- | 7 || ৭ সাত |Saat (Sāta) |- | 8|| ৮ আট |Aatt (Āṭa) |- | 9 || ৯ নয় |Noy (Naẏa) |- | 10 || ১০ দশ |Dosh (Daśa) |} {{subpage navbar}} {{CourseCat}} [[Category:Numbers]] fhjgq0eubzqsj4wabgdi47usiyctwyz 0 241565 2844154 2761944 2026-11-03T09:52:14Z Jtneill 10242 /* Motivation */ + points from Google class doc 2844154 wikitext text/x-wiki {{/Banner}} ==2025== ===Motivation=== * The use of autonomy in helping or guiding others. I have noticed that when I give my kids more choices it has resulted in better results for all of us * Something that unexpectedly resonated with me was fixed-growth mindsets! Having found I lean towards the former, finding out that it and its associated performance goals tend to be unhealthy and unsustainable. Trying to move toward more of a growth-mindset now! * Part of my book chapter found that a lot of traditionally used motivators can actually be really detrimental to motivation, or at least, optimal motivation, because they diminish autonomy. So having autonomy is really key for the best kind of motivation. * That people need autonomy so much that we will engage in psychological reactance to reassert that autonomy even when it may not be in our best, long-term interests * I’ve also tried to be more intrinsically motivated about tasks, and “find the fun”, so to speak. * Framing all behaviours as some degree of motivation was really interesting and thinking about “why am I doing this behaviour?”. Not just the super energetic ones that are typically framed as being motivated, but also things like “why am I motivated to just stay in bed this morning?”. * The neuroanatomy and neurotransmitters involved in motivation. * Understanding how being neurodivergent can impact motivation. * Exploring psychological needs * Learning about mindsets - cognitive patterns or mental frameworks and how mindsets influence thinking, behaviours and feelings * This unit solidified for me that motivation and the factors/ triggers that cause someone to feel motivated or demotivated are so unique to a person such as in the beginning of the unit we discussed two different reasons for people to go for a run in the morning for person 1 is training for a marathon or running to keep fit while person 2 is going for a run to catch up with friends or as a social outing. * Learning motivation helps us to understand what we want, why we want and how. * Understanding how relatedness, autonomy and competence can be applied in an organisational or work setting to achieve a certain outcome. * Learning about implicit motives. It made me realise that motives aren’t always something we are consciously aware of, and it was quite fascinating noticing that I am predominantly driven by affiliation. * This class reaffirmed a lot of the things I knew about mindsets and how they can shape our mental health * Motivation is both an internal individual process as well as an external influenced process * Freud’s dual instinct theory - Eros (life instinct) and Thanatos (death drive) - positive vs destructive coping mechanisms to unconscious instincts * I really enjoyed this unit as a whole especially thought it was interesting to learn about the nucleus accumbens role in motivation. * For me it was learning about self determination theory and applying it into the way I motivate myself and others. In particular the ideas around relatedness, autonomy and competence. * * ===Emotion=== ===Other=== ==See also== * [[/Previous years/]] [[Category:Motivation and emotion]] 4r7p79jfhipnye71lx7oam81vbfyjz9 2844155 2844154 2026-11-03T09:53:14Z Jtneill 10242 /* Emotion */ + points from Google class doc 2844155 wikitext text/x-wiki {{/Banner}} ==2025== ===Motivation=== * The use of autonomy in helping or guiding others. I have noticed that when I give my kids more choices it has resulted in better results for all of us * Something that unexpectedly resonated with me was fixed-growth mindsets! Having found I lean towards the former, finding out that it and its associated performance goals tend to be unhealthy and unsustainable. Trying to move toward more of a growth-mindset now! * Part of my book chapter found that a lot of traditionally used motivators can actually be really detrimental to motivation, or at least, optimal motivation, because they diminish autonomy. So having autonomy is really key for the best kind of motivation. * That people need autonomy so much that we will engage in psychological reactance to reassert that autonomy even when it may not be in our best, long-term interests * I’ve also tried to be more intrinsically motivated about tasks, and “find the fun”, so to speak. * Framing all behaviours as some degree of motivation was really interesting and thinking about “why am I doing this behaviour?”. Not just the super energetic ones that are typically framed as being motivated, but also things like “why am I motivated to just stay in bed this morning?”. * The neuroanatomy and neurotransmitters involved in motivation. * Understanding how being neurodivergent can impact motivation. * Exploring psychological needs * Learning about mindsets - cognitive patterns or mental frameworks and how mindsets influence thinking, behaviours and feelings * This unit solidified for me that motivation and the factors/ triggers that cause someone to feel motivated or demotivated are so unique to a person such as in the beginning of the unit we discussed two different reasons for people to go for a run in the morning for person 1 is training for a marathon or running to keep fit while person 2 is going for a run to catch up with friends or as a social outing. * Learning motivation helps us to understand what we want, why we want and how. * Understanding how relatedness, autonomy and competence can be applied in an organisational or work setting to achieve a certain outcome. * Learning about implicit motives. It made me realise that motives aren’t always something we are consciously aware of, and it was quite fascinating noticing that I am predominantly driven by affiliation. * This class reaffirmed a lot of the things I knew about mindsets and how they can shape our mental health * Motivation is both an internal individual process as well as an external influenced process * Freud’s dual instinct theory - Eros (life instinct) and Thanatos (death drive) - positive vs destructive coping mechanisms to unconscious instincts * I really enjoyed this unit as a whole especially thought it was interesting to learn about the nucleus accumbens role in motivation. * For me it was learning about self determination theory and applying it into the way I motivate myself and others. In particular the ideas around relatedness, autonomy and competence. * * ===Emotion=== * I was surprised that interest was an emotion. I had never thought of interest in this way but now I find I utilise it in daily activities. For example in revision I try and find little topics of interest that encourage more reading/learning * Learning about the vast amount of emotions and understanding the differences between some and how to distinguish between them * Learning about emotional intelligence and it being divided by awareness and management and self vs. social or relational. * It was interesting learning different definitions for both basic and complex emotions as typically I wouldn’t describe them as the provided definition eg. anger, however when applied to the context of motivations of behaviour, it all makes sense and clicks better for me * Understanding how being neurodivergent can impact emotions. * Emotions are not as easy to categorise as I thought * I learnt not to internalise emotions or label them too quickly * All emotions have value and purpose and can be leveraged to help with motivational endeavours. Anxiety can be reappraised as excitement. * Learning from my book chapter helped me bridge emotion and motivation together and look at their relationship differently * It was interesting learning about the distinction between basic, cognitively complex, and self-conscious emotions. * How each emotion has a communicative function, which is recognisable across cultures. * Human emotion is a very tricky thing. Emotion is the most interesting part of Psychology I would like to dig deeper into. * Emotions can be motivators * Schadenfruede - evil joy :) * The power of synthetic happiness is astonishing! ===Other=== ==See also== * [[/Previous years/]] [[Category:Motivation and emotion]] dsc5vky36g9khzr73dc8vb68s5mfpjq 2844156 2844155 2026-11-03T09:53:43Z Jtneill 10242 /* Other */ + points from Google class doc 2844156 wikitext text/x-wiki {{/Banner}} ==2025== ===Motivation=== * The use of autonomy in helping or guiding others. I have noticed that when I give my kids more choices it has resulted in better results for all of us * Something that unexpectedly resonated with me was fixed-growth mindsets! Having found I lean towards the former, finding out that it and its associated performance goals tend to be unhealthy and unsustainable. Trying to move toward more of a growth-mindset now! * Part of my book chapter found that a lot of traditionally used motivators can actually be really detrimental to motivation, or at least, optimal motivation, because they diminish autonomy. So having autonomy is really key for the best kind of motivation. * That people need autonomy so much that we will engage in psychological reactance to reassert that autonomy even when it may not be in our best, long-term interests * I’ve also tried to be more intrinsically motivated about tasks, and “find the fun”, so to speak. * Framing all behaviours as some degree of motivation was really interesting and thinking about “why am I doing this behaviour?”. Not just the super energetic ones that are typically framed as being motivated, but also things like “why am I motivated to just stay in bed this morning?”. * The neuroanatomy and neurotransmitters involved in motivation. * Understanding how being neurodivergent can impact motivation. * Exploring psychological needs * Learning about mindsets - cognitive patterns or mental frameworks and how mindsets influence thinking, behaviours and feelings * This unit solidified for me that motivation and the factors/ triggers that cause someone to feel motivated or demotivated are so unique to a person such as in the beginning of the unit we discussed two different reasons for people to go for a run in the morning for person 1 is training for a marathon or running to keep fit while person 2 is going for a run to catch up with friends or as a social outing. * Learning motivation helps us to understand what we want, why we want and how. * Understanding how relatedness, autonomy and competence can be applied in an organisational or work setting to achieve a certain outcome. * Learning about implicit motives. It made me realise that motives aren’t always something we are consciously aware of, and it was quite fascinating noticing that I am predominantly driven by affiliation. * This class reaffirmed a lot of the things I knew about mindsets and how they can shape our mental health * Motivation is both an internal individual process as well as an external influenced process * Freud’s dual instinct theory - Eros (life instinct) and Thanatos (death drive) - positive vs destructive coping mechanisms to unconscious instincts * I really enjoyed this unit as a whole especially thought it was interesting to learn about the nucleus accumbens role in motivation. * For me it was learning about self determination theory and applying it into the way I motivate myself and others. In particular the ideas around relatedness, autonomy and competence. * * ===Emotion=== * I was surprised that interest was an emotion. I had never thought of interest in this way but now I find I utilise it in daily activities. For example in revision I try and find little topics of interest that encourage more reading/learning * Learning about the vast amount of emotions and understanding the differences between some and how to distinguish between them * Learning about emotional intelligence and it being divided by awareness and management and self vs. social or relational. * It was interesting learning different definitions for both basic and complex emotions as typically I wouldn’t describe them as the provided definition eg. anger, however when applied to the context of motivations of behaviour, it all makes sense and clicks better for me * Understanding how being neurodivergent can impact emotions. * Emotions are not as easy to categorise as I thought * I learnt not to internalise emotions or label them too quickly * All emotions have value and purpose and can be leveraged to help with motivational endeavours. Anxiety can be reappraised as excitement. * Learning from my book chapter helped me bridge emotion and motivation together and look at their relationship differently * It was interesting learning about the distinction between basic, cognitively complex, and self-conscious emotions. * How each emotion has a communicative function, which is recognisable across cultures. * Human emotion is a very tricky thing. Emotion is the most interesting part of Psychology I would like to dig deeper into. * Emotions can be motivators * Schadenfruede - evil joy :) * The power of synthetic happiness is astonishing! ===Other=== * Really liked the time perspective tutorial and this is something I’ve been considering in my day to day life since * I’m interested in learning more about positive psychology. I think it’s so important to not merely look at people from a deficit point of view. Don’t just look at what’s wrong, look at what’s going right! * I have found positive psychology really interesting. I * Learning how to use wikiversity and learning so much from the bookchapters * I would be interested to learn more about the time perspective as it was something I didn’t know much about and have tried applying it to everyday life and shifting my perspectives of things to have an equal balance as to which point in time i live in more than another. * Learning about the emphasis of meaning and purpose in positive psychology was a really nice reminder of finding a way to make things feel meaningful where I can. * I found it really interesting taking the activities and questionnaires from tutorials and applying it to my friends, specifically the time constraints we looked at. Getting them to complete them and predict their scores based on what I know of their personality was really fun and helped to apply it and see it among people I know. * Learning the importance of hobbies in facilitating well-being really stuck with me. * The influence of personality on these aspects was something I wasn’t aware of prior * Understanding how the time perspective can influence positive and negative outcomes in a person's life, as well as how positive psychology provides a more holistic view of psychology, which is often not discussed in much depth. * Self-actualising isn’t just becoming really good at something. It can expand and reflect outwards for the greater good, and help others reach their optimal being. ==See also== * [[/Previous years/]] [[Category:Motivation and emotion]] pm46c2z60xoqcfpldc7lzwmxkiyo88e 2844159 2844156 2026-11-03T09:59:34Z Jtneill 10242 Move 2025 to subpage 2844159 wikitext text/x-wiki {{/Banner}} ==2026== ===Motivation=== ===Emotion=== ===Other=== ==See also== * [[/Previous years/]] [[Category:Motivation and emotion]] huwc9c17zzj3cwbxpdssgiunxtbwsab 0 264724 2844005 2843394 2026-11-02T19:18:42Z Scogdill 1331941 2844005 wikitext text/x-wiki {{Short description|Dress worn by Queen Victoria at her wedding to Prince Albert in 1840}} = Sandbox = Page to draft revisions for Wikipedia articles. For Gwladys Robinson, see Gwladys Lowther Robinson, [[Social Victorians/People/Ripon|Marchioness of Ripon]] and, earlier, [[Social Victorians/People/Lowther|Countess of Lonsdale]] ==References== {{reflist|2}} [[Category:1840 works]] [[Category:Royal wedding dresses|Victoria Queen]] [[Category:1840s fashion]] [[Category:British royal attire]] [[Category:Dresses in the Royal Collection of the United Kingdom|Victoria, Wedding]] [[Category:Diamond Jubilee of Queen Victoria]] = Victorian fashion = == Notes and Quotes == * Women followed Paris fashion in Europe and America, but London tailors led the way for men’s fashion. The Industrial Revolution actually impacted men’s fashion more than women’s. * Sewing machines in factories began mass producing cheap clothing for the largest group in the population, working class men. * Ready-made clothing affected the social structure, the economy, and gender differentiation. * Paper patterns, introduced in fashion magazines that targeted women, along with domestic sewing machines had more impact on women’s fashion than men’s. Haute couture was still bespoke (being individually constructed by tailors or dressmakers or fashion houses like Maison Worth). Sewing Machines and Clothes Before the industrial revolution wealthy people paid others to sew their clothes by hand and those who were of much lower economic status did all the manual labor necessary to feed and clothe the rich minority. The working class had to make all their own clothes. Sewing, however, came at the end of the process.  They began with raw wool cut from the backs of sheep or gathering flax. Only the wealthy could afford to buy fabric. They followed processes for cleaning and spinning the wool into thread and the processes of soaking and pulling out the fibrous strips to make into thread for linen fabric. After they had thread, it was tied to a loom so it could be woven into cloth. Then it had to be cut or torn into pieces. For many families scissors were a luxury. And the last step is sewing the pieces together to create a garment. Machines could not only sew, they could process the raw materials into thread and weave it into fabric much  quicker. Regarding clothing, working class men and their families benefitted greatly from ready-made clothes. Manufacturers marketed to them because they were the largest class. For example,<blockquote>–Men’s shirts, made by machine, take 3 hours and 10 minutes each to do; by hand, 15 hours and 20 minutes; a lady’s dress, by machine, takes 12 hours 4 minutes; by hand 20 hours 35 minutes; . . . a common frock-coat, by the machine occupies 17 hours 20 minutes; by the hand, 27 hours and 40 minutes: a linen vest, by machine, 3 hours 44 minutes; by hand 7 hours 18 minutes; a fine overcoat, by machine, 28 hours 13 minutes; by hand 31 hours 20 minutes; . . . <ref>Mitchell, Rebecca, ed. "22. 'Progress of the Sewing Machine,' ''Bow Bells'' (19 April 1865), p. 283." Section 6: Production and Industry. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.</ref></blockquote> === Industrial Revolution and Technological Advancements === * the railroad * mass production of fabrics for working-class men's clothing ** Jacquard looms * sewing machines * aniline dyes, including black === Expanding Middle Class === * gender roles getting more clearly defined and rigid (referring back to page overview) === French Revolution === Sarah Gharmallah Alzahrani and Safia Abdelaziz Saroukh (https://www.researchgate.net/profile/Safia-Saroukh/publication/385099194_The_Semiotic_Dimension_of_Men's_Fashion_in_Modern_Eras/links/671686fbd796f96b8ec4f90e/The-Semiotic-Dimension-of-Mens-Fashion-in-Modern-Eras.pdf):<blockquote>The 19th century started with a fashion landscape that was changing dramatically and rapidly from the styles of a generation earlier. The French Revolution brought fashions that had been emerging since the 1780s to the forefront. Neoclassicism now defined fashion as both men and women taking inspiration from classical antiquity. For women, the high-waisted silhouette in lightweight muslin was the dominant style, while fashionable men looked to the tailors of Britain for a new, refined look [17]. [17]  Franklin, H. (Aug 18, 2020). Published on Jun 25, 2020, Retrieved: <nowiki>https://fashionhistory.fitnyc.edu/1800-1809/</nowiki> 11/11/2023. Edited. ...</blockquote>Brent Shannon:<blockquote>"Costume," wrote Max Beerbohm in 1896, "enables us to classify any 'professional man' at a glance, be he lawyer, leech or who not" (24–25). A man's profession and class were read by his jacket, his hat, what he rode in, and how he carried himself. "Perhaps there is a tendency among Englishmen to judge a man too much by the shape of his hat or the kind of collar he wears," conduct author John Wanamaker confessed; "But one must remember that in England if you ''wear'' the wrong thing, you will probably ''do'' the wrong thing, and generally ''be'' the wrong thing" (1).¹¹ Such assertions were predicated on the powerful Victorian conviction that outward appearance reflected inner qualities.<ref>{{Cite book|title=The Cut of His Coat: Men, Dress, and Consumer Culture in Britain, 1860–1914|last=Shannon|first=Brent Alan|publisher=Ohio University Press|year=2006|location=Athens, Ohio}}</ref> (148)</blockquote> ==Women's fashion== == Men's fashion == (Two intros: one to Victorian Fashion, giving background for both women's and men's fashions; two to the decades for men's fashion) Among the influences on men's styles: * differentiation of genders (domestic ideology, ideology of the spheres) * the "Great Male Renunciation," the French Revolution and English tailoring * industrial revolution: mass production of fabrics and articles of clothing, especially for working-class men * sports and country clothing * Albert Edward, Prince of Wales, later King Edward VII, influence went beyond England; tailors led by him led fashion for the upper classes * then later, cult of youth * military uniforms * comfort and convenience * evolving definitions of gender, both femininity and masculinity evolved as concepts. The changing definitions of masculinity affected men's clothing. For the growing and rising middle classes, gender roles became more and more rigid around the concept of the separate spheres, the public for men and the private sphere for women. Cultural sources for men's clothing and men's fashion: industrial revolution; expanding middle class; technological advancements; an association in the culture between outward appearance and inward nature; French Revolution; Albert Edward, Prince of Wales Citation for Victorian Hell: <ref name=":29">{{Cite web|url=https://victorianhell.substack.com/p/victorian-mens-fashion-history-and|title=Victorian Men’s Fashion History and Clothing Guide|last="It's Monty, Actually"|date=31 October 2025|website=Victorian Hell|access-date=7 September 2026}}</ref> === Frock coats === * Apostol Mănescu, politician, wearing a frock coat, butterfly tie, black waistcoat with button chain and light coloured pants, drainpipe type. Circa 1866. Photo by Franz Duschek.jpg (1866)[[File:Apostol Mănescu, politician, wearing a frock coat, butterfly tie, black waistcoat with button chain and light coloured pants, drainpipe type. Circa 1866. Photo by Franz Duschek.jpg|thumb|Romanian (?) Politician, c. 1866, ill-fitting but probably handmade suit|center]] Full-length portrait of two young men wearing frock coats with sashes and pistols at their waists LCCN2006689564.tif (right)[[File:Full-length portrait of two young men wearing frock coats with sashes and pistols at their waists LCCN2006689564.jpg|thumb|1860, American, young men with frock coats, performers??|center]] * George Augustus Henry Sala.jpg (Mathew Brady, 1855–1865) * [[File:George Augustus Henry Sala.jpg|center|thumb|George Augustus Sala, 1855–1865, LoC]] ==== Dress and Conduct Manuals, Histories, Victorian Theory, Catalogs ==== # [Baker, William Henry.]  ''Clothes Dictionary''. Chicago, Ill.: Cahn, Wampold, 1901. https://commons.wikimedia.org/wiki/File:Clothes_dictionary;_(IA_clothesdictionar00bake).pdf. # [[c:File:Hint's_about_men's_dress,_right_principles_economically_applied_(IA_hintsaboutmensdr00barr).pdf|Hint's about men's dress, right principles economically applied (IA hintsaboutmensdr00barr).pdf]] (book, conduct manual, American, 1888) #[https://upload.wikimedia.org/wikipedia/commons/6/69/Fashion-_the_power_that_influences_the_world_%28IA_fashionpowerthat00foxg%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original Fashion- the power that influences the world (IA fashionpowerthat00foxg).pdf] (1871) (American, incl court dress, Victorian fashion theory, livery, "White, as it is the color of day, is expressive to us of the cheerfulness or gaiety which the return of day brings." [145], "Gentlemen's Furnishing and Goods," "Head and Foot Covering," Index) # [https://upload.wikimedia.org/wikipedia/commons/a/ae/How_to_behave_and_how_to_amuse._A_handy_manual_of_etiquette_and_parlor_games_%28IA_howtobehavehowto00sand%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original How to behave and how to amuse. A handy manual of etiquette and parlor games (IA howtobehavehowto00sand).pdf] (1895; American; Men's Dress, p. 23; besides conduct and dress, it has parlor games) # [https://upload.wikimedia.org/wikipedia/commons/8/8e/The_etiquette_of_men%27s_dress_%28IA_etiquetteofmensd00cluc%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original The etiquette of men's dress (IA etiquetteofmensd00cluc).pdf] (1888; American; some ads; anthology of individually written articles; lots of sports, outdoor events; search for "frock" — cutaway vs Prince Albert) #[https://upload.wikimedia.org/wikipedia/commons/e/e9/S.W._Silver_%26_Co.%27s_handbook_for_transatlantic_travellers_.._%28IA_swsilvercoshandb00silv%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original S.W. Silver & Co.'s handbook for transatlantic travellers .. (IA swsilvercoshandb00silv).pdf] (1883; American; lots of ads from SW Silver, with prices; has blank pages where ppl can write "Introductions" and "Engagements," but this is not a conduct book, it's travel guide / catalog to London) #[https://upload.wikimedia.org/wikipedia/commons/f/f6/How_we_go_to_those_who_cannot_come_to_us.._%28IA_howwegotothosewh00roge%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original How we go to those who cannot come to us.. (IA howwegotothosewh00roge).pdf] (Roger, Peet & Co.; American; catalog, spring 1891; pp. 11, clerical frock; 7 "Prince Alberts" double-breasted frocks; p. 23 shirts as underwear; has a thing on fitting and mail order; bathing suits, collars and cuffs; tennis shoes) == Victorian Men's Fashion (Intro to the Men's Fashion Section) == Museums and other historical sites have plenty of women’s clothes from the nineteenth century. There is great variety in both outer wear and underwear. “Men’s garments of the Victorian period have survived in far less quantity than women’s.” Buck p. 185 More waistcoats have survived than any other garment. The hardest garment to find are trousers. Very few survived the nineteenth century. Buck p. 185 By 1850 trousers dominated the fashion for men; the trouser straps under the shoe had disappeared; and a fly front had replaced the fall-front. Buck 185 It took most of the nineteenth century for men’s fashion to evolve into the coat, trousers and vest (waistcoat) that dominated the twentieth century and appears to be unchanging in the twenty-first century. “Men’s clothing, like women’s, was specific to each occasion and type of activity during the late nineteenth century.” Payne2, p. 518 For example, formal wear for social events was different from the casual, everyday wear for upper middle class and aristocratic men. Victorians also dressed in ensembles for hunting and riding (included knickerbockers–breeches that had a fuller cut). Casual and looser clothing was worn when participating in sports. Many different cuts of coats were available, like the Norfolk jacket, the morning coat, or the sack coat. “Tailored wool garments , originally inspired by the clothing of English country gentry, formed the backbone of men’s wardrobes in the early nineteenth century.” Men’s clothing began to change from styles that were colorful and flamboyant in the eighteenth century to styles that emphasized cut over color. By the end of the century men’s clothing had settled into a kind of uniform with coats and trousers usually made of the same fabric, often with a vest or waistcoat of the same fabric. The uniform included a shirt with a stiff collar, a necktie and very little jewelry. It took the entire nineteenth century to change to a rather static style that is still worn today. Ready-made clothing affected the social structure, the economy, and the gender differentiation. David Kutcha traces the origin of the 3-piece men's suit to King Charles II of England, who may be said to be wearing a coat, pants and a waistcoat, but otherwise is not wearing a "suit" we — or the Victorians — would recognize as such: <blockquote>... since 1666, male gentility has been associated with modesty and plainness in dress. Eschewing fashion as an increasingly feminized realm Charles II's vest inaugurated a new and essentially modern era of masculine aesthetics, one that reversed a long-held association between elaborate display and high social status. Manly thrift now displayed elite status.<ref>{{Cite book|title=The Three-Piece Suit and Modern Masculinity, England 1550–1850|last=Kutcha|first=David|publisher=University of California Press|year=2002|location=Berkeley and Los Angeles}}</ref> (2)</blockquote>So it's really Cromwell who should get credit?? The French Revolution caused both men and women to change stylistic direction radically. For one generation, stylish women wore simple white muslin dresses without foundation garments underneath. The changes in men's fashion was more lasting: men turned away from the ornamentation and ruffles of the ''ancien régime'', choosing instead understated and subtle tailoring to indicate good taste and wealth. Poole "comfort and convenience" — trousers, sack coats, collars and cuffs without a shirt, wool rather than silks and satins; knee breeches constantly riding up men as leaders of fashion, in spite of our sense that men didn't do fashion in the way that women did, were more serious than that === Dandy Stuff === Beau Brummell: "Men began to aspire to fine cutting, tailoring, and perfect fit in their clothes, flawless grooming and manners in themselves. The Englishman George Bryan (Beau) Brummell deserves much credit for the ideal of meticulous masculine appearance."<ref name=":25">{{Cite book|title=The History of Costume: From Ancient Mesopotamia Through the Twentieth Century|last=Payne|first=Blanche|last2=Winakor|first2=Geitel|last3=Farrell-Beck|first3=Jane|publisher=Addison-Wesley Longman|year=1992|isbn=0-06-047141-7|edition=2nd|location=New York, New York}}</ref> (458) "From 1796 to 1816, the Beau set the pattern for cleanliness and liberal use of starch." (459) === Gender Ideologies === '''Victorian gender ideologies''' relegated men to the "public sphere" and women to the "domestic sphere." (??) More than that “the cult of youth . . . the secularization of sport and the influence of modern warfare in generating men’s fashions, the tendency to prefer modern ideas of comfort and convenience, and the transformation of fashion knowledge and fashion urbanism from print to hyperreality”<ref name=":26">McNeil, Peter. "Men's Fashion: 1800–2022." Chapter 22. ''The Routledge History of Fashion and Dress, 1800 to the Present''. Routledge, 2024. https://opus.lib.uts.edu.au/bitstream/10453/182707/2/Men%27s%20Fashion%20200822_24_12_20_09_29_44.pdf DOI: 10.4324/9780429295607-27.</ref> (McNeil 1 or 411) changed the perceptions of masculinity and femininity. === Class === The growing middle class involved among other things more and more jobs and careers for young men as clerks, office workers, they were junior, subordinate, and they were commuting on the railroads from suburbs. Carolyn Kirby:<blockquote>In western Europe the fashion for plain dark suits coincided with the rise of the affluent middle-classes in a world where the pace of industrialisation and the globalisation of trade was accelerating as never before. The sharp, dark business suit became the last word in male power-dressing. And so it remains to this day.<ref name=":27">{{Cite web|url=https://historiamag.com/invent-masculine-fashion/|title=The invention of masculine fashion|last=Kirby|first=Carolyn|date=3 December 2025|website=Historia: Magazine of the Historical Writers' Association|access-date=25 August 2026}}</ref></blockquote> === Bertie === Albert Edward, Prince of Wales was very concerned with fashion and authoritative about it, with a very specific eye to small details. He was increasingly influential in men's fashion from the last third of the 19th and into the 20th century, and many noticed what he did and copied him.<blockquote>When that great lover of pleasure, Edward VII, visited Marienbad incognito as the Duke of Lancaster, he was followed by tailors from Paris, Budapest, Vienna, and Berlin who photographed him and took notes about his clothes. Edward VII introduced many [423–424] novelties into men’s fashion. ... He is also credited with making fashionable the creased trouser in 1909 (his groom dried them with a board weight after heavy rain, resulting in the line), turned-up cuff trouser (after hitching his trouser bottoms at a dirty racing track) and, as his girth grew, undoing the bottom button of his waistcoat."<ref name=":26" /> (424)</blockquote>Cowles<blockquote>The Prince was not just ‘a swell’. In the jargon of the day he was ‘a heavy swell’, and apparently there was a world of difference between the two terms. A swell was a rich young aristocrat who lived in extreme comfort; but a heavy swell added showmanship to the comfort and lived in a stylish luxury that even the French were obliged to envy. And of course the heavy swell was the acme of sartorial elegance. The Prince did not mind changing his dress half a dozen times a day. He loved clothes, and since whatever he chose to wear became the prevailing fashion overnight, he soon was regarded as an expert on the subject. His tailor-in-chief made a fortune. For many years he patronised a Mr. Poole. He discovered this gentleman by accident. He went to the theatre one night to see a well-known actor by the name of Fecher playing ‘Robert Macaire’. As an impecunious adventurer [129–130] Fechter was obliged to wear a coat that was torn and dirty, but Bertie’s expert eye noticed the elegant cut. At the end of the performance he asked Fechter for the name of his tailor, and Mr. Poole’s future was assured. The Prince had so many clothes he could never travel with less than two valets; and two more valets were left at home cleaning, brushing and pressing his vast wardrobe. There were suits and coats for every variation of every climate the world over. There were over a hundred pieces of headgear; and since Bertie was an honorary admiral and an honorary general of most of the countries of Europe, there was an entire room devoted to uniforms, sashes, epaulettes, belts, buckles, swords, feathers and other regalia. ... Men’s clothes became of such importance that new [130–131] shops sprang up like mushrooms in Savile Row, Clifford Street and Bond Street. Most of the Prince’s innovations were inspired by comfort and convenience. He altered the cut of the evening dress waistcoat, he shortened the tails on the tail coat, he left his frock coat open (due to an increasing girth), he introduced the black homburg, and he attended race meetings, not in the frock coat hitherto ''de regueur'' but in tweeds. He tried having his trousers creased down the sides rather than the front and back, in order to hide his bandy legs, but this idea did not catch on, and he soon discarded it himself. But the prince was not the only arbiter of men’s fashions. The band of  "heavy swells" who followed his lead gave him plenty of competition. Lord Raglan and Lord Petersham invented coats which are still named after them. Lord Dupplin the dinner jacket and Lord Cardigan the button-up sweater. But Lord Hardwicke made the most spectacular contribution. Men’s silk hats were made of beaver which was left in its original rough, shaggy state. Lord Hardwicke polished his hat until he could see his face in it, and consequently was known as "Glossy Top". He is responsible for the top hat as we know it today.<ref name=":28">{{Cite book|title=Gay monarch, the life and pleasures of Edward VII|last=Cowles|first=Virginia|publisher=Harper|year=1956|location=New York, New York|archive-url=https://archive.org/details/gaymonarchlifepl0000cowl/}}</ref> (129–131)</blockquote> === Men’s Clothing 1830–1900 (Intro to the Decades) === “Tailored wool garments , originally inspired by the clothing of English country gentry, formed the backbone of men’s wardrobes in the early nineteenth century.”<ref name=":25" /> (458) Men’s clothing began to change from styles that were colorful and flamboyant in the eighteenth century to styles that emphasized cut over color. By the end of the century men’s clothing had settled into a kind of uniform with coats and trousers usually made of the same fabric, often with a vest or waistcoat of the same fabric. The uniform included a shirt with a stiff collar, a necktie and very little jewelry. It took the entire nineteenth century to change to a rather static style that is still worn today. (this is middle-class men for day wear?) The silhouette for men sometimes followed the same line as women’s clothing on the top half of their bodies. For example, as women’s sleeves grew into the huge leg-of-mutton fullness, the shoulders of men’s coats also had added fabric to make a pouf at the shoulder, thus emphasizing a broad shouldered look across the front. Men’s silhouettes changed subtly without the exaggerated fullness of women’s skirts or the raising and lowering of the neckline. Once men began to wear trousers, they did not go back to formal knee breeches (except in European courts or for court wear for Victoria). The line of men’s garments changed in smaller increments than women’s. Coats and trousers might fit tighter in some decades and looser in others. They were slim, then boxy, flared, then straight and evolved into the contemporary line we see on men today. Men's fashions in the 19th century have been less researched than women's fashions, in part because of fewer artifacts of what men wore were saved in museums for study. Photography, the first century that we have actual images (rather than artists' interpretations) of what people wore. ==== Court and Formal Dress ==== Livery, working class Formal dress was separate attire, used for specified occasions, varies by decade ==== Shirts ==== Shirts and collars separated: "by 1827 detachable collars became available."<ref name=":25" /> (477) Shirts were considered more like underwear early in the century but by the end of the 1800s, they were becoming the outside garments popular in the twentieth and twenty-first centuries. Neck treatments (and shirt cuffs) Shirts were treated as underwear early in the century but by the end of the 1800s, they were becoming the outside garments popular in the twentieth and twenty-first centuries. ==== Coats and Jackets ==== The 1901 ''Clothes Dictionary'' says coats were "The principal garment of men's attire."<ref name=":30" />{{rp|10}} Several styles of coats were available to fashion-conscious aristocratic and upper-middle-class men. Men’s suits in this period were tailor made, fitted to one specific man. The factories, however, knew that the vast majority of men were working class and began to manufacture and sell ready-made shirts, coats and trousers. More clothes were available for working men and they cost less. The cheaper work clothes enabled upper-working-class men to own more than one set of clothes and to have more formal clothes for church or other events. Ready-made shirts were also available in generic sizes, small, medium and large. Generic sizing speeded up the mass production process and, since little of the shirts could be seen, they did not need custom fitting. Almost all of the coats Victorian men wore could be single or double breasted, with a couple of exceptions like morning coats and Norfolk jackets. On more formal occasions upper-class men wore a '''cutaway morning coat''' with light trousers during the daytime, and a dark '''tail coat''' and trousers in the evening. (define these terms) Morning coats was a general term for the jacket part of "full dress" or formal daytime wear, distinguished from formal evening wear. Most morning coats were cutaways, but the shape or cut is not the distinguishing feature: the bottom was curved or tailed, so the coat was short in front, it did not have a waist seam, single-breasted. Worn over a waistcoat and shirt. According to Payne, "During the early 1820s tailcoats were the most popular type of coat."<ref name=":25" />{{rp|474}} "Tailcoats gave way to skirted frock coats by the 1840s."<ref name=":25" /> (491) Tailcoats came down as far as the waist and then had tails (or "claw hammer" tails) attached at the waist in the back. Single-breasted, could be worn buttoned or unbuttoned in the front, over a waistcoat and shirt. Nowadays, the only tailcoats are tuxedos, a term not in use until the 1880s, and named for Tuxedo Park, a village in New York populated by the elite. Originally only the white coat, ''tuxedo'' gradually came to mean the entire suit. ==== Frock Coats ==== [[File:Frock coat MET CI38.23.39 F.jpeg|alt=Old-fashioned coat hanging on a mannequin|thumb|Frock coat, 1840, American]] Frock coats were introduced in the 1820s and worn throughout the nineteenth century with only minor changes to the line or fit. Sleeves were full at the top and waists were tight, creating an hourglass form. Initially frock coats were tightly fitted and closed with buttons. The more tailored frock coats had a waist seam and a flared skirt almost to the knees. The rate of fashion change notable in women's fashion was true for men's clothing as well. The frock coat persisted for a century, but it was in a state of constant (but subtle) evolution, with collars and lapels, coat length, the waist seam, the fullness of the coat itself as well as the sleeve, color, patterns like stripes or plaids. A Prince Albert is a double-breasted, very well-tailored, form-fitting frock coat.<ref name=":30">[Baker, William Henry.] ''Clothes Dictionary''. Chicago, Ill.: Cahn, Wampold, 1901. https://commons.wikimedia.org/wiki/File:Clothes_dictionary;_(IA_clothesdictionar00bake).pdf.</ref>{{rp|33}} Gentlemen, clergy, military Men around the world and especially in the British Empire wore frock coats: images and descriptions exist of men wearing frock coats in India<ref>Woodyatt, Nigel Gresley. ''Under Ten Viceroys; The Reminiscences of a Gurkha''. London: H. Jenkins Limited, 1922. Pp. 31, 107, 1774.</ref>, Iran (Persia at that time)<ref>Sparroy, Wilfrid. ''Persian children of the royal family; the narrative of an English tutor at the court of H. I. H. Zillus-Sultán, G.C.S.I''. London and New York, J. Lane, 1902. Pp. 11, 29, 30.[https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf].</ref>, Japan<ref>Watson, W. Petrie. ''Japan, Aspects and Destinies''. New York, Dutton: 1904. P. 7, 19, 31.</ref>, Egypt and Turkey<ref>Butler, Alfred Joshua. ''Court Life in Egypt''. London: Chapman and Hall, 1887. P. 7.</ref>, ... Payne: "In the 1720s, Englishmen adopted another coat [than the redingote]: a straighter, functional style that originated among the working classes. Made of plain woollen cloth in practical colors such as brown, it had no pleats, narrow sleeves, a small flat collar, and a minimum of decoration. This became known as the frock coat and was useful for riding, hunting, and overseeing one's lands." (402) ==== Trousers ==== Working-class men had been wearing trousers (or "trowsers") increasingly since the 16th century in Europe,('''citation''') but after the French revolution the middle and upper classes adopted them because "the safest dress was that of the working man, ... who did not wear knee breeches, but rather trousers."<ref name=":26" /> (416) Breeches: Men’s bifurcated garment that ended just below the knee Trousers (or long pants): men’s bifurcated garment that went down to the ankles or foot The most sweeping change in men’s fashion before 1820 was in the length of their pants. Trousers that went from waist to foot were so much more comfortable and popular than knee breeches that they replaced what had dominated men’s wear since the sixteenth century. And trousers have been worn for over two centuries now. They were originally tailored but began to be mass produced. Although the aristocracy were required to wear knee breeches at court, they wore trousers more than they wore breeches, and they could afford to have their suits and shirts hand sewn and individually. Working class men had to contend with small, medium, and large from mass production. This bifurcated garment was modified in specific places, like the length, the fullness over the hips or the width of the leg. In summer trousers were lighter in color — "grayish blue, aqua and pearl gray." (Payne 2, 505) In winter trousers mixed darker colors like "steel gray and brown, black and brown, and black and green." (Payne 505) More on summer colors, winter colors? [Find quote about QV's men's court dress and the men being so cold] Lewandowski: "First worn by the early horse-riding steppe people, introduced to Western world by Italians in 16th century. They were accepted as informal dress in the Directoire (1790–1815 C.E.). They became a part of semi-formal dress around 1815 and became worn commonly after 1830."<ref name=":31" /> (29 ==== Waistcoats ==== For most of the century, the waist-length sleeveless coat that men wore over their shirt and under their suit coat was called a waistcoat. By the 1880s, everyday waistcoats were called vests, and what was called a waistcoat was worn with formal, evening attire like tailcoats and tuxedos. Necklines rose and fell and collars and lapels came and went over the century, but by the end of the century, formal waistcoats had low necklines and lapels.<ref name=":25" /> (528) Many more waistcoats exist as artifacts in collections than vests because they were special, perhaps, and so they were saved.[[File:Waistcoat MET CI60.22.55 F.jpg|thumb|Men's waistcoat, 1850s]] Over the course of the century the waistcoat evolved from ornate and decorative to something simpler, whose fabric matched the suit coat and trousers — the vest, in fact. Victorian men put their creativity, color and specialty fabrics into their waistcoats. At the beginning of the century men's waistcoats were made from specialty fabrics, like velvet or brocade or a patterned silk. Early in the century, stylish men wore two vivid waistcoats at the same time.<ref name=":25" /> (474) An elaborate waistcoat might have a colorful, patterned base fabric as well as a floral ribbon embellishment (for example, see right). Long after coats and trousers were made from the same dark wool, waistcoats continued to be colorful, making them the focus of the ensemble. By the end of the 19th century, the plain style came to dominate men’s fashion (and continues to dominate it to this day). When waistcoats became vests, they became part of the plain style. * cc. 1830, dark brown or black w faces, gold buttons, standing collar: File:Waistcoat MET 38.68 F.jpg * c. 1830, beige and white woven fabric with rose buds stamped or embroidered??: File:Waistcoat MET 19.181.4 F.jpg * 1830–1835, dotted and floral stamped fabric, brown and tan: File:Waistcoat MET 1976.254.1.jpg * * c. 1840, flowered fabric: File:Waistcoat MET 1979.346.45 F.jpg * c. 1840, red, jacquard-like, double-breasted: File:Waistcoat MET 1979.346.47 F.jpg * 1840s, white brocade: File:Waistcoat MET 1980.132.8 F.jpg * 1840s, dark blue, red, green? stamped?: File:Waistcoat MET 1976.254.1.jpg * Mid 1840s, brown, embroidered: File:Waistcoat MET 24.160.3 F.jpg * late 1840s, embossed white with hash thingies: File:Waistcoat MET 1978.88.9 F.jpg * 1845 waistcoat made from blue Chinese silk damask: File:1845 waistcoat made from blue Chinese silk damask.jpg * 1845–1855, white brocade wedding waistcoat, piping on the edges: File:Waistcoat MET 1986.373.3 F.jpg * '''1849–1860''', black w gold lining, ribbon: File:Waistcoat MET 1978.477.25 F.jpg * * 1850, print of St. James's Park: File:Print (BM 1880,1113.2351).jpg * 1850s, white and blue, w blue floral design stamped on white-and-grey stamped on white fabric: File:Waistcoat MET 1978.477.28 F.jpg *c. 1853, brown and tan plaid, with red squares: File:Waistcoat MET 1984.90.2 F.jpg *1854: Fashion plates, groups of ppl: File:Paris, New York & Philadelphia fashions for spring and summer 1854, published & sold by F. Mahan, no. 211 Chestnut Street, Philadelphia. LCCN2003655770.jpg *c. 1855, French fashion plate, frock coats, waistcoats, etc.: File:Album z ilustracjami do czasopism z moda francuska, niemiecka i angielska. ca 1855 (9109644).jpg * 1857, cream, printed flowers on check, watch pocket: File:Waistcoat MET 1978.88.8 F.jpg * * c. 1860, dark floral? fabric with machine embroidery?: File:Waistcoat MET 1983.479.13 F.jpg * * late 1870s–early 1880s, gold, woven: File:Waistcoat MET 1979.34.1 F.jpg [[File:Homosexualitywilde.jpg|thumb|Oscar Wilde and Bosie Douglas, straw boater and derby, 1894]] ==== Overcoats ==== Overcoats or surcoats were, as their name suggests, coats worn for warmth and for protection from rain and snow on top of the other articles of clothng, including suit coats and frock coats. Overcoats became a significant fashion element and many styles were seen throughout the century. Victorians used the term overcoat or great coat, and they also specified what kind of overcoat: Chesterfield coat, Inverness coat, macintosh (or mackintosh), paletot, redingote, surtout, Ulster coat Surtout: "An overcoat cut in the style of a frock coat."<ref name=":30" /> (39) Lewandowski says something more generic, "term referring to any man's long cloak or coat."<ref name=":31">{{Cite book|title=The Complete Costume Dictionary|last=Lewandowski|first=Elizabeth J.|publisher=Scarecrow Press|year=2011}}</ref> (282) ===== Redingote ===== Redingotes had a rolled collar, sleeves with fullness at the shoulders and upper arm and narrow at the elbows to the wrist. It was fitted at the waist and reached mid-calf. existed various forms in the 18th century. Women's redingote 1865–1890, in France, to cover the bustle. (Lewandowski, 247–248) Payne: "The greatcoat or redingote evolved into an overcoat ..., but the overcoat lacked the pleats and large cuffs of the redingote." (402) high, double collar, standing collar, single- or double-breasted. "from riding coat" (Payne 402) Rebecca Mitchell: "full-length overcoat, usually open at the front and belted, popular in varying designs through the eighteenth and nineteenth centuries."<ref name=":32">Mitchell, Rebecca, ed. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.</ref> (527 or 565) ===== Chesterfield Coat ===== "Chesterfield coats were single-breasted, with fly fronts [regular buttons and button holes, as opposed to fall front, in which a panel of cloth covers the button holes and the placket covers the closure completely] and velvet collars."<ref name=":25" />  Payne2 p. 519 They were one kind of overcoat, named for George Stanhope, 6th Earl of Chesterfield. 1840s. Signature velvet collar. "Man's fitted overcoat with hidden buttons and velvet collar"<ref name=":31" /> (62) "fitted men’s knee-length overcoat, narrower than the frock coat and without a waist seam, popular from the end of the nineteenth century."<ref name=":32" /> (519 of 565) ===== Ulster Coat ===== ulster overcoat for travel. "Man's single- or double-breasted overcoat with waist belt. Initially had hood, but by 1870s was detachable. In 1875, the ticket pocket was added on the left sleeve above the cuff."<ref name=":31" /> (304) "Travel called for an ulster — a long, loose, heavy overccoat — or for an Inverness coat with attached cape."<ref name=":25" /> (527) "Heavy tweed ccoats for travel such as the ulster had half-belts and box pleating in back."<ref name=":25" /> (551) In 1872, George Darwin (Charles's son), in analyzing men's clothing as evolutionary behavior — like coloration for mating among birds — in the natural world, says, of trousers and Ulster coats:<blockquote>In both cases a form yields to a succeeding form, which is better adapted to the then surrounding conditions; thus, when it ceased to be requisite that men in active life should be ready to ride at any moment, and when riding had for some time ceased to be the ordinary method of travelling, knee breeches and boots yielded to trousers. The "Ulster Coat", now so much in vogue, is evidently largely fostered by railway travelling, and could hardly have flourished in the last century, when men either rode or travelled in coaches, where there was no spare room for any very bulky garment.<ref>Mitchell, Rebecca, ed. "3. 'George H. Darwin, ‘Development in Dress', ''Macmillan’s Magazine'' 26 (September 1872), pp. 410–16." Section 1: Fashion Theory in the Nineteenth Century. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.</ref> (125 of 565)</blockquote>"men’s overcoat with attached waist-length cape, popular in the nineteenth century."<ref name=":32" /> (530 of 565) ===== Inverness Coat ===== an Inverness coat with up to five cape collars and a high curved collar. Lewandowski: "(1840–1865 C.E.) to 20th century. Worn after 1858, man's large, loose, knee-length overcoat with fitted collar and long cape."<ref name=":31" /> (143) An Inverness coat had a high standing collar as well as up to 5 capes over a knee-length, loose overcoat.<ref name=":31" /> (143) In the 1880s, the Inverness was sometimes sleeveless, so the capes were not merely decorative. "men’s overcoat, with long, full, open sleeves, giving a cape-like appearance from the front; worn from the mid-nineteenth century to the early twentieth century as formal wear."<ref name=":32" /> (523 of 565) ===== Greatcoat ===== "large overcoat, commonly made of wool" (Lewandowski, 124) ===== Macintosh ===== "Introduced in 1836, short loose overcoat made of Mackintosh's patent India-rubber cloth. Came in drab or dark green."<ref name=":31" /> (181) "waterproofed overcoat, named after its inventor; now used generically to refer to raincoat."<ref name=":32" /> (524 of 565) ===== Paletot ===== "short, loose overcoat for men or women."<ref name=":32" /> (526 of 565) <ref name=":24" />Kay, Fiona and Neil R. Storey, ''Victorian Fashions for Women'', Pen & Sword History, Yorkshire - Philadelphia, 2022 ==== Hats ==== The upper-class continued to wear top hats, and [[bowler hats]] (known in America as a derby) were worn by the working class. 1830s This class of men also wore round cloth caps, often seen on male children. The straw boater was the most popular hat by the end of the century and was worn by all of the classes. Men wore [[top hats]], with wide brims in sunny weather. 1830s - 1900 Top hats briefly became the very tall "stovepipe" shape, but a variety of other hat shapes were popular. Homburgs, deerstalkers and “Trilbies” were also popular. Buck, p. 198-199 (George de Maurier novel ''Trilby'') Hats were ''de rigueur'' for all Victorians, all classes, all ages, all genders. ==== Shoes ==== [[File:Men's fashion silhouette of 1837.jpg|thumb|Men's coats, 1837]] [[File:Lev Wittgenstein by F. Kruger.JPG|thumb|Lev Wittgenstein by F. Kruger, 1836]] === 1830s and 1840s === For men's fashion, the 1830s and 1840s were dominated by a defined waist, lean limbs and tall hats. Working-class men had been wearing trousers for some time, but by the 1830s and '40s, middle- and upper-class men generally wore frock coats and trousers for casual or less formal attire in the city. In the country, what they wore depended on what they were doing — hunting, riding, shooting and so onen[[File:Man's Frock Coat LACMA M.2007.211.61 (1 of 4).jpg|alt=Male mannequin wearing a light-colored thigh-length coat similar to a suit jacket and a top hat|thumb|Men's frock coat, c. 1845]] [[File:Stockholms mode-journal- Tidskrift för den eleganta werlden 1847, illustration nr 2.jpg|thumb|Stockholm's fashion magazine, 1847]]Some men wore corsets during this time, so the waist was nipped in and the skirt of the frock coat flared out. The sleeves of their coats and jackets were slightly puffed at the shoulders. The trouser legs were straight with a strap under the instep. The figure on the left in the fashion plate from ''Modes de Paris Journal des Tailleurs'' (right) illustrates this silhouette very clearly as well as the frock coat, the strap under the instep, and the puffed shoulders. ==== Coats and Jackets ==== Frock coats had become fashionable for day wear in the 1820s<ref name=":25" />{{rp|474}} and were worn throughout the century. In the 1830s frock coats were tightly fitted and closed with buttons. The more tailored frock coats had a waist seam and a flared skirt almost to the knees. The main changes in frock coats in these two decades were in the colors and the neck treatments (collar and lapels). Initially frock coats were not necessarily black. They could be colorful — like "plum, mulberry, brown, bright blue, and bottle green" — with "contrasting velvet collars."<ref name=":25" />{{rp|474}} The truly bright, saturated colors made possible by synthetic dyes, however, were a couple of decades away still and were never used for frock coats. Middle- and upper-class men limited their use of tailcoats to evening wear and more formal occasions. (In the 1847 Stockholm fashion magazine, below right, the figure in the center is wearing a tailcoat.) A cutaway morning coat was worn with light trousers for formal events during the daytime, and a dark tail coat and trousers were worn in the evening. ==== Shirts ==== Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[cravats]] or neck ties. Before they were mass produced, shirts made of finer linen or cotton were tailored for upper and wealthy middle-class men. Depending on occupation, working-class men's shirts were made from a rougher, home-woven linen or cotton, often shaped loosely like a smock and sometimes without a front closure. Shirts were sometimes as long as knee-length. ==== Trousers ==== Men had adopted trousers over knee breeches for semi-formal dress "around 1815"<ref name=":31" /> (298) beginning after the French Revolution.<ref name=":26" /> (416) They were wearing trousers both for formal and semi-formal dress by the 1830s. Initially trousers "fit quite snugly ..., but in the 1840s they became looser."<ref name=":25" /> (477) Upper-class and aristocratic men still wore knee breeches at court and loose-fitting breeches (in the form of knickerbockers) for country wear and some sports, like riding and golf. Generally, however, they wore trousers more than they wore breeches. Men flocked to them once trousers became a stylish option because the looser fit was warmer in cold weather and more comfortable than the tight-fitting knee breeches from earlier in the century. Ready-made men's clothing was available in the US early, but by the 1840s ready-made suits were making an impact in Europe and the UK,<ref name=":26" /> (419–420) especially outside the upper classes. Aristocrats and upper middle class could afford the well-tailored cut and fit of individually hand-sewn trousers and coats. ==== Waistcoats ==== Waistcoats were both single- or double-breasted, with shawl or notched collars, and might be finished in double points at the lowered waist. Collars gradually lowered over the course of the two decades. Often the colorful vests were decorated with fur or velvet collars. “Waistcoats remained colorful, decorative items in men’s wardrobes after the rest of their costumes had become plain.” (Payne 2 460-462) <nowiki>:</nowiki> "During the 1820s and 1830s, men sometimes wore two waistcoats, in combinations like white velvet over rose-and-gold brocade."<ref name=":25" /> (474) ==== "Overcoats ==== Men wore a variety of topcoats or overcoats. Surtouts were overcoats built like long frock coats, like the one worn by Lev Wittgenstein in the portrait (above right). The redingote was most popular in the 1830s and 1840s. ==== Original Text ==== During the [[1840s in fashion|1840s]], men wore tight-fitting, calf length [[frock coat]]s and a [[waistcoat]] or vest. Sleeves were full at the top and waists were tight, creating an hourglass form. For more formal occasions, a cutaway morning coat was worn with light trousers during the daytime, and a dark tail coat and trousers was worn in the evening. Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[Cravat (early)|cravat]]s or neck ties. Men wore [[top hat]]s, with wide brims in sunny weather. === 1850s === The 1850s revealed several industrial inventions that significantly impacted clothing. First, the Industrial Revolution began (and still does today) with machines to weave fabric and machines to sew fabrics. Manufacturers quickly learned that the majority of the population was working class. Theirs were the simplest, least trimmed garments made in the nineteenth century. Second, mass production of shirts, trousers and coats changed how most people shopped for clothes. The aristocracy and upper middle class folks could still have their clothes made to specifically fit their bodies. Manufacturers knew that they could not mass produce clothing for each individual’s body. They invented generic sizing in small, medium, and large. They could, therefore, produce many more items of clothing with generic sizing. Third, Isaac Singer patented and began to manufacture home or domestic sewing machines (treadle machines) enabling more form-fitted '''dresses''' to be constructed for an individual’s specific shape and size. The fourth item to impact fashion and clothing was the invention of the paper pattern. These were introduced in fashion magazines targeting women. Almost all women were taught to sew by hand, whether they were aristocrats or working class. Sewing machines enormously speeded up the '''process of constructing from hours to minutes'''. Finally, aniline dyes made colors brighter and longer lasting. Prints and stripes, checks and plaids, fabrics had much more vibrant color with aniline dyes. The Industrial Revolution impacted the working class and changed their lives the most. children's clothes? Women made men's clothing too According to Judith Flanders, working-class men's clothes could be colorful. For example drivers of hansom cabs () dressed more up-scale, and drivers of hackneys () did not.<blockquote>While hackney drivers were also considered to be stereotypically shabby, hansom-cab drivers were generally represented as smartly dressed. A print in 1850 showed a driver in a snappy brown coat instead of the coachman’s heavy multiple-caped outfit, pale green striped trousers, short boots and top hat, the [167–168] reins held daintily in his gloved hands. Both cab and coach drivers wore top hats, but cabbies of a sporting bent later switched to bowlers, and in summer donned bright checked outfits.<ref name=":23">{{Cite book|title=The Victorian City: Everyday Life in Dickens' London|last=Flanders|first=Judith|publisher=Thomas Dunne Books|year=2012|location=New York, New York}}</ref> (167–168 [of 972])</blockquote> A variety of silhouettes was seen in the 1850s. Trousers were long and slim with a strap under the instep of the shoe. ==== Coats and Jackets ==== "Although the frock coat was worn throughout the century, it lost its dominant position in the 1850s, when the morning coat began to replace it."<ref name=":29" /> A jacket shorter than the frock coat or morning coat that had visible pockets began to be worn in about 1850. Cutaway tail coats were still reserved for formal occasions. All of these coats could be single or double breasted. They were called “ditto suits” by those who wore them because the coat, trousers, and waistcoat were all made from the same fabric. There were morning coats and cutaway morning coats; there were Norfolk jackets and ditto suits. The most formal were the cutaway tail coats for evening wear and balls; and the least formal were the boxy shaped sack coats. By the 1860s, it had become the most popular casual coat. ==== Shirts ==== During the [[1850s in fashion|1850s]], men started wearing shirts with high upstanding or turnover [[collar (clothing)|collars]] and [[necktie#Four-in-hand|four-in-hand necktie]]s tied in a bow, or tied in a knot with the pointed ends sticking out like "wings." ==== Trousers ==== Trousers had fly fronts and breeches were used for formal functions and knickers (short pants fuller cut than breeches) for horseback riding. Men's choice to wear trousers is an instance of the importance of comfort and convenience outweighing style and presentation, a value that increased in importance through the 20th century. ==== Waistcoats ==== Waistcoats remained the one garment that could be brightly colored and copiously trimmed on exotic fabrics. Waistcoats show that men were concerned with decorative elements in their ensemble. ==== Notes & Quotes ==== * transition from frock coats to ditto suits, 1850s (Payne, 463) ==== Hats ==== The upper-class continued to wear top hats, and [[bowler hat]]s were worn by the working class. Caps ==== Overcoats ==== Overcoats were cut fuller with wider sleeves and no extra fullness at the shoulder. Unlike the redingote, the overcoats had straight seams to the knee length. Many had wide lapels that reached below the waist. Working class men and their families benefitted more from the industrial revolution in regards to clothing.  Manufacturers marketed to them because because they were the largest class. [[File:Gazette of Fashion, February 1865, RP-P-2009-2675.jpg|thumb|1865, Frock coats, etc. (Paris)]] [[File:Gazette of Fashion, October 1864, RP-P-2009-2673.jpg|thumb|1864, Frock coats, etc., Paris]] [[File:Broadway fashions, New York LCCN2003677917.jpg|thumb|Butterick, Spring-Summer 1866, Broadway Fashions]] [[File:Four young men wearing frock coats - DPLA - 4706a21c9267c45034d51c8fc5309251 (page 1).jpg|thumb|1860s, working class?, ill fitting, frock coats, trousers, shirts, vest, derby hats]] === 1860s === [[File:Corporal frock coat 1862 med.jpg|thumb|Corporal frock coat 1862, 9-botton military]]The 1860s saw fewer inventions that influenced or changed the way people were acquiring clothes. But the changes begun by mass production continued with the working class. Little changed for the higher classes, except maybe paper patterns and sewing machines. The different shape and size in clothing continued to provide slightly different silhouettes for men in the upper and middle classes. As women's clothes grew more complex in cut and style, men's clothes became simpler in cut and style. ==== Coats and Jackets ==== Individualized fittings continued for upper class and wealthy middle class clients. (Working class as well as middle and upper class.) Working class was well over 60% of the population. For the largest class, ready-made garments were ideal for men and women who were working class. For the two upper classes morning coats still led the way but frock coats were still in demand and worn in business and more casual situations. Leisure wear like knickers and ditto suits became more popular in the 1860s. Haute couture wanted every occasion or event to have special clothing. In some ways men dressed more alike (coat, trousers, waistcoat) and in some ways there was more variety in the cut of their coats and in the manufactured clothing they could buy. Many upper class men still had their shirts custom made. Frock coats were shortened to knee-length and were worn for business, while the mid-thigh length [[sack coat]] slowly displaced the frock coat for less-formal occasions, with the overall effect of a looser silhouette. The Prince of Wales: "For the countryside such as at Sandringham, he permitted an informal dress code. The Henry Poole ledger marked as 'HRH 1865' is for an evening coat without tails, the first 'dinner jacket.'"<ref name=":26" /> (424) ==== Shirts ==== Ready-made shirts rarely were seen since shirts were more underwear than outerwear. Factories began to make and sell detachable collars and cuffs. It was more convenient to replace collars and cuffs each day than the time it took a servant (working class) to wash and starch enough to look like the celluloid many of those collars were made of. ==== Trousers ==== Trousers were cut full and boxy during most of the 1850s and 1860s. The strap under the shoe disappeared and trouser legs reached the back of the heel in length. Knickers were still worn for hunting or riding and breeches were still ''de rigueur'' at court. ==== Waistcoats ==== Waistcoats remained decorative and worn with most of the coats that were popular. The young “dandies” wore the most colorful and decorated waistcoats. These “dandies” gave their highest priority to their appearance. They were the avant garde of haute couture for men. ==== Necks Treatments ==== In the [[1860s in fashion|1860s]], men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin. ==== Hats ==== Top hats briefly became the very tall "stovepipe" shape, but a variety of other hat shapes were popular. ==== Overcoats ==== The single-breasted redingote continued in popularity among the aristocracy and upper middle-class.  These coats were often “in colors such as plum, mulberry, brown, bright blue, and bottle green . . . [with] contrasting velvet collars.” Payne2 p. 474 [[File:Mens Coats 1872 Fashion Plate.jpg|thumb|upright|Drawing of Victorian men 1870s]] [[File:Ashurst frock coat.jpg|thumb|Ashurst Frock coat]] [[File:Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat; Standing (fbc266c2feca43f6ad5b44e031635a8d).tif|alt=Old photograph of a man wearing a dark thigh-length coat similar to a suit coat with light-colored trousers|thumb|Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat, c. 1872]] [[File:Caillebotte Hugot.jpg|thumb|Portrait of Paul Hugot, 1878]] [[File:Oscar Wilde (1854-1900), by Hills & Saunders, Rugby & Oxford 3 april 1876.jpg|thumb|Oscar Wilde, 1876, ditto suit]] [[File:FrockCoat.jpg|thumb|English frock coat, 1870s]] === 1870s === To correct and prevent errors being made in court dress, in 1875 the Lord Chamberlain published ''Dress Worn by Gentleman at Her Majesty's Court'', "a summary of regulations for court uniform and dress."<ref>{{Cite book|url=https://www.google.com/books/edition/Dress_worn_by_Gentlemen_at_Her_Majesty_s/pvrbQCXq0MEC?hl=en|title=Dress worn by Gentlemen at Her Majesty's Court|last=Britain)|first=Victoria (Queen of Great|date=1875|language=en}}</ref> This is the kind of thing Bertie really cared about. Define court dress. A portly, boxy silhouette was sought for most of the 1870s. Older men whose stomachs had spread outward and thus looked prosperous to the Victorians were the ideal for men’s high fashion. Sack coats and trousers were cut a little wider in the legs. Corduroy for trousers and muslin for shirts was often used in working class clothing. ==== Coats and Jackets ==== There were minor changes to some of the coats Victorian men wore. Frock coats were shortened to the knee. Sack coats (or ditto coats — the forerunner of the suits worn by men in the western cultures of the twentieth and twenty-first centuries) were found to be so comfortable that most men of the upper and middle classes had a ditto suit. Men still wore morning coats and the aristocracy still wore cutaway coats with trousers to parties and balls but cutaway coats with knee breeches at court. ==== Shirts ==== Although ready-made shirts were available, the upper and middle classes still wore bespoke clothing hand-made, with better fabric and an excellent fit but stitched on machines. The suits and shirts made for the working class were of cheaper fabrics. ==== Trousers ==== Most men were wearing trousers  all of the time (except at court) and have worn them ever since. Trousers had button flies by the 1870s which most men found to be much more convenient than laces. Trousers did not “ride up” the leg in the way that knee breeches did when dancing or sitting. If a man did not have fashionable legs, they could be obscured behind trousers. ==== Waistcoats ==== White waistcoats began to be worn by men. Black were also worn when mourning. Changes in the waistcoat could be seen but were very subtle. For example, collars were lowered as the neckline grew deeper and deeper in front, revealing more of the fancy vest. ==== Overcoats ==== A longer version of the standard overcoat, “sometimes with fur collar, cuffs, and lining” was often worn for warmth. Payne2 p. 519 ==== Original Text ==== During the [[1870s in fashion|1870s]], three-piece suits grew in popularity along with patterned fabrics for shirts. Neckties were the four-in-hand and, later, the [[Ascot tie]]s. A narrow ribbon tie was an alternative for tropical climates, especially in the Americas. Both frock coats and sack coats became shorter and more form fitting. Flat straw boaters were worn when boating. [[File:Elliott & Fry10a.jpg|thumb|William Gladstone, 1887, morning coat]] [[File:British politicians gambling at Ascot Wellcome V0050378.jpg|thumb|British Politicians Gambling at Ascot, 1884]] [[File:Scribner's magazine (1887) (14595087238).jpg|thumb|1887, types of American servants]] === 1880s === The silhouette narrowed and straightened. That is, the line went straight down without the flare of skirts on the coats. This smooth line had few decorations. Men wore hardly any jewelry. They might have a boutonniere or a ring.  Suits were not  decorated. All coats were straighter and shorter. Depending on the time of day, the Prince of Wales might change his clothes (thus, his silhouette) up to six times per day,<ref name=":28" /> (129) and some aristocrats and upper-class men would follow his lead. ==== Coats and Jackets ==== Although the frock coat was worn throughout the nineteenth century, the single fabric sack suit was worn for casual wear and was beginning to be seen as a businessman’s suit.  Morning coats were still in use for some events as well as casual occasions. The modern tuxedo or Cowes jacket appeared and was worn for formal occasions. The tuxedo is still worn as formal dress today. ==== Shirts ==== Shirts were still custom made for the rich and the working class bought ready-made shirts. The shirt remained basically the same but it could handle the different neck treatments.  Even mass produced shirts could have buttons or button holes for detachable collars and cuffs. ==== Trousers ==== Trouser legs began to decrease in width as the coat became narrower and emphasized strong vertical lines. ==== Waistcoats ==== During the 1880s waistcoats were called vests [is it regional?] and much of their bold trim and colorful fabric disappeared and vests were constructed from the same fabric as the coat and trousers. The business suit for men developed in the 1880s and was firmly established in the 1890s. ==== Overcoats ==== Versions of topcoats increased, adding an ulster overcoat for travel and an Inverness coat with up to five cape collars and a high curved collar. Double breasted topcoats were popular and some were knee-length and some calf-length. ==== Original Text ==== During the [[1880s in fashion|1880s]], formal evening dress remained a dark tail coat and trousers with a dark waistcoat, a white bow tie, and a shirt with a winged collar. In mid-decade, the dinner jacket or [[tuxedo]], was used in more relaxed formal occasions. The [[Norfolk jacket]] and tweed or woolen breeches were used for rugged outdoor pursuits such as shooting. Knee-length topcoats, often with contrasting velvet or fur collars, and calf-length overcoats were worn in winter. Men's shoes had higher heels and a narrow toe. [[File:American fashions - The Jno. J. Mitchell Co. Litho., New York. LCCN2006676665.jpg|thumb|Fashion plate from the Jno. J. Mitchell Co., New York, 1899]] [[File:Alfred Milner Vanity Fair 1897-04-15.jpg|thumb|Alfred Milner, 1897, Leslie Ward ("Spy"), for Vanity Fair]] [[File:Nobl-08.jpg|thumb|Sherlock Holmes, 1892, Sidney Paget]] [[File:PUK1894page380rector poor.jpg|thumb|Rector and Drinker, 1894]] === 1890s === The dominant silhouette for men became the sack suit in the 1890s and while changes in the sack have been subtle, it remains the dominant silhouette today. ==== Coats and Jackets ==== “Sacks became ''the'' favored style of suit coat,”(Payne2, p. 532) especially among the younger men. Frock coats were associated with older men along with the cutaway coat. There were many more choices for sports wear and leisure wear than ever before. Norfolk jackets were often worn while hunting. Most informal styles could be purchased ready made. The rich still had their clothing custom made by tailors but even working class men could purchase most ready-made versions of the coats, trousers, and vests. ==== Shirts ==== Shirts could be made of patterned fabric as well as solid colors and white.  Shirts were beginning to be acceptable as outer wear especially when worn as sporting clothing. ==== Trousers ==== Early in the 1890s, “slim, straight trousers were news.” (Payne 2, p. 532) Trousers could be plaid or checked as well as the same fabric as the coat and vest. ==== Waistcoats ==== Fancy fabrics and bold trim were not seen much as the more subdued vest dominated the fashion scene. ==== Neck Treatments ==== In the 1860s, men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin. ==== Overcoats ==== “Topcoats included fingertip-length styles made of twill covert cloth. Single-breasted, fly-front overcoats were fashionable too. Roomy overcoats with fur linings and collars warmed the wealthy. Inverness overcoats remained in vogue.” Payne2, p.532 ==== Original Text ==== Starting from the [[1890s in fashion|1890s]], the [[blazer]] was introduced, and was worn for sports, sailing, and other casual activities.<ref>{{cite web|last=Landow|first=George|url=http://www.victorianweb.org/art/costume/90s/2.html|title=Men's informal sporting dress, late 1880s and '90s}}</ref> Throughout much of the Victorian era most men wore fairly short hair. This was often accompanied by various forms of facial hair including moustaches, side-burns, and full beards. A clean-shaven face did not come back into fashion until the end of the 1880s and early 1890s.<ref>{{cite web|url=http://www.victorianweb.org/art/costume/nunn21.html|title=Victorian Men's Fashions, 1850–1900: Hair}}</ref> Distinguishing what men really wore from what was marketed to them in periodicals and advertisements is difficult, as reliable records do not exist.<ref name="shannon597">{{cite journal|last=Shannon|first=Brent|title=Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914|journal=Victorian Studies|year=2004|volume=46|issue=4|pages=597–630|doi=10.1353/vic.2005.0022}}</ref> === Notes === *Men's suits buttoned higher up than today (Payne, 467) * Norfolk jackets and sack suits (Payne, 471) * formal attire, tuxedos with tails, cutaways (Payne, 469) * Keith Middlemas (https://archive.org/details/storyoffiesta00huxf/page/200/mode/2up?q=fashion) *Brent Shannon. "Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914." Victorian Studies 46, no. 4 (Summer 2004): 597–630. == Hats and headwear == [[File:Ford.madox.brown.last.emma.study.jpg|thumb|''Emma Hill'' by [[Ford Madox Brown]] (1853), a woman wearing a later version of the [[poke bonnet]]]] [[File:Hoed,_objectnr_KA_1237.tif|left|thumb|Perched bonnet style of the early 1870s.]] Hats were crucial to a respectable appearance for both men and women. === Men's Hats === The top hat, for example, was standard formal wear for upper- and middle-class men.[Payne] According to Blanche Payne, "The high top hat, usually black or dark gray, had reached its characteristic shape by 1798 and dominated the entire nineteenth century." (457–58) Although top hats were the dominant hat in the 19th century, other hats became popular for working classes and lower income middle class. “The style of an individual’s hat varied, depending on fashion and their social position, as well as their profession or chosen activity..” ''Goodman 53 of 460'' Other hats that became necessary and popular include the Derby, the straw Boater, and a flat cap with a short brim. In many cases the class, work activity and income could be determined by what kind of hat was on the head of the wearer. For some men, a hat supporting a particular sport or team was important. The Derby or Bowler hat was designed by William and Thomas Bowler, brother shopkeepers in 1849. ''(Goodman 55 of 460)'' It cost less than a top hat but  lasted longer and was soon worn by middle class bankers and clerks. Straw boaters were worn by the aristocracy for casual events and working class factory workers and agricultural laborers. By 1901 working class men had changed their preference to the flat caps which became the most popular hat for the workers. Headdress for men was an essential part of dress for the entire period of the Victorian age, from the 1830s through the end of the century. Judith Flanders describes the hats worn by men in London, <blockquote>It is difficult to bear in mind the importance of hats as not only markers of class and income, but also as indicators of respectability. [509–519] [George Augustus] Sala commented that "every" man throughout the history of the world "must, necessarily and habitually, wear some kind of covering to his head". Postmen wore hats, small children wore hats, field labourers and market gardeners wore hats, cricketers, skaters — all sportsmen — wore hats. It was, self-evidently, impossible to go outdoors without one. ... Those in professional occupations wore pot hats, as did clerks and all those with pretensions to middle-class status. Even doctors' delivery boys wore battered hand-me-down pot hats: "the nap rusty, the band a mournful strip of tarnished lace; but still a Hat", which "stamps him as being associated, in however slender a manner, with a learned profession". Cloth caps were for labourers, for costers and for boys. ... Artisans wore caps made out of paper, which they folded [510–511] themselves and so could easily replace as they became dirty.<ref name=":23" /> (509–511 [of 972]) </blockquote>A pot-hat is a kind of derby or bowler in men's hats. (Lewandowski, 237) ==== Original Text ==== Hats were crucial to a respectable appearance for both men and women. The top hat, for example, was standard formal wear for upper- and middle-class men.<ref name=":4">{{Cite book |last=Steele |first=Valerie |url=https://archive.org/details/fashioneroticism0000stee |title=Victorian Fashion. Fashion and Eroticism: Ideals of Feminine Beauty from the Victorian Era to the Jazz Age |publisher=Oxford University Press |year=1985 |isbn=978-0-19-503530-8 |pages=[https://archive.org/details/fashioneroticism0000stee/page/51 51]–84 |url-access=registration}}</ref> For women, the styles of hats changed over time and were designed to match their outfits. === Women's Hats === For a discussion of the history of plumes and feathers, see [[Social Victorians/Victorian Things#Ostrich Plumes and Prince of Wales's Feathers|Ostrich Plumes and Prince of Wales's Feathers in ''Victorian Things'']]. ==== Original Wikipedia Text ==== During the early Victorian decades, hats were modest in size and design, straw and fabric bonnets being the popular choice. [[Poke bonnet]]s, which had been worn during the late [[Regency period]], had high, small crowns and brims that grew larger until the 1830s, when the face of a woman wearing a poke bonnet could only be seen directly from the front. They had rounded brims, echoing the rounded form of the bell-shaped hoop skirts. Bonnets shrunk at the end of the 1860s and moved to a perched position in the early 1870s as hairstyles grew in scale and intricacy. This led to the popularization of hats, which became the headwear of choice for the remainder of the Victorian era.<ref name="g4223">{{cite book |last=Cunnington |first=Cecil Willett |title=English Women's Clothing in the Nineteenth Century |date=1990-05-01 |publisher=Courier Corporation |isbn=0-486-26323-1 |publication-place=New York |page=}}</ref> [[File:The_London_and_Paris_ladies'_magazine_(Apr_1885)_03.png|thumb|Flower pot style hat of 1885.]] The 1880s saw a hat inspired by the top hat for women known as the flowerpot hat, and the 1890s saw the popularity of the boater. The hats of the late Victorian era were covered with elaborate creations of silk flowers, ribbons, and above all, exotic plumes; hats sometimes included entire exotic birds that had been stuffed. Many of these plumes came from birds in the Florida everglades, which were nearly made entirely extinct by overhunting. By 1899, early environmentalists like [[Adeline Knapp]] were engaged in efforts to curtail the hunting for plumes. By 1900, more than five million birds a year were being slaughtered, and nearly 95 per cent of Florida's shore birds had been killed by [[Plume hunting|plume hunter]]s.<ref>{{cite web|title=Everglades National Park|url=https://www.pbs.org/nationalparks/parks/everglades/|archive-url=https://web.archive.org/web/20090927085907/http://www.pbs.org/nationalparks/parks/everglades/|url-status=dead|archive-date=27 September 2009|publisher=PBS|access-date=7 November 2011}}</ref> == Shoes == The women's shoes of the early Victorian period were narrow and heelless, in black or white satin. By 1850s and 1860s, they were slightly broader with a low heel and made of leather or cloth. Ankle-length laced or buttoned boots were also popular. From the 1870s to the twentieth century, heels grew higher and toes more pointed. Low-cut pumps were worn for the evening.<ref name=":4" /> == Cosmetics == [[Victorian-era cosmetics]] were typically less obvious than ours. However, small amounts of pale face powder or powdered blush were widely used.<ref>{{Cite book |last=Goodman |first=Ruth |title=How to be a Victorian |date=2014 |publisher=Penguin Books |isbn=978-0-670-92136-2 |location=London}}</ref> Tints were sometimes added to face creams. Some cosmetics contained toxic or caustic ingredients like lead, mercury, ammonia, and arsenic {{Citation needed|date=October 2025}}. Hair color ==Mourning black== {{See also |Mourning stationery}} [[File:The royal children in mourning Mar 1862.jpg|thumb|Victoria's five daughters (Alice, Helena, Beatrice, Victoria and Louise), photographed wearing mourning black beneath a bust of their late father, Prince Albert (1862)]] [[File:Mourning dress MET 50.40.3a-b front CP4.jpg|alt=Black Victorian mourning dress|thumb|Mourning Dress, 1894–95]] In Britain, black is the colour traditionally associated with mourning for the dead. The customs and etiquette expected of men, and especially women, were rigid but evolving during much of the Victorian era. The expectations depended on a complex hierarchy of close or distant relationship with the deceased. (Davidoff) The closer the relationship, the longer the mourning period and the wearing of black. The wearing of full black was known as First Mourning, which had its own expected attire, including fabrics, and an expected duration of 4 to 18 months. Following the initial period of First Mourning, the mourner would progress to Second Mourning, a transition period of wearing less black, which was followed by Ordinary Mourning, and then Half-mourning. Some of these stages of mourning were shortened or skipped completely if the mourner's relationship to the deceased was more distant. Half-mourning was a transition period when black was replaced by acceptable colours such as lavender and mauve, possibly considered acceptable transition colours because of the tradition of [[Church of England]] (and [[Catholic Church|Catholic]]) clergy wearing lavender or mauve [[Stole (vestment)|stoles]] for funeral services, to represent the [[Passion (Christianity)|Passion of Christ]].<ref>{{cite web|title=The Colors of the Church Year|url=http://fullhomelydivinity.org/articles/colors.htm|publisher=Consortium of Country Churches|access-date=6 November 2011|archive-date=13 November 2011|archive-url=https://web.archive.org/web/20111113075214/http://fullhomelydivinity.org/articles/colors.htm|url-status=dead}}</ref> The mourning dress worn by Queen Victoria (below, right) "shows the traditional touches of mourning attire, which she wore from the death of her husband, Prince Albert (1819–1861), until her own death."<ref>{{Cite web|url=https://www.metmuseum.org/art/collection/search/155839?&searchField=All&sortBy=Relevance&deptids=8&ft=queen+victoria&offset=0&rpp=20&amp;pos=2|title=Mourning Dress, 1894–95|last=The Metropolitan Museum of Art|date=7 September 2019|website=The Metropolitan Museum of Art|access-date=7 September 2019}}</ref> Dating from 1894–95, Queen Victoria wore this dress as a result of the death of the eldest son of the Prince and Princess of Wales, Eddy, in line to the throne. === Norms for mourning=== ''Manners and Rules of Good Society, or, Solecisms to be Avoided'' (London, Frederick Warne & Co., 1887) gives clear instructions, such as the following:<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–83}}</ref> {| class="wikitable" |- ! Relationship to deceased !! First mourning !! Second mourning !! Ordinary mourning !! Half-mourning |- | Wife for husband || 1-year, 1-month; [[bombazine]] fabric covered with [[Crape|crepe]]; [[widow's cap]], [[lawn cuff]]s, collars || 6 months: less crepe || 6 months: no crepe, silk or wool replaces bombazine; in last 3 months jet jewellery and ribbons can be added || 6 months: colours permitted are grey, lavender, mauve, and black-and-grey |- | Daughter for parent || 6 months: black with black or white crepe (for young girls); no linen cuffs and collars; no jewellery for first 2 months || 4 months: less crepe || – || 2 months as above |- | Wife for husband's parents || 18 months in black bombazine with crepe || – || 3 months in black || 3 months as above |- | Parent for son- or daughter-in-law's parent || – Black armband in representation of someone lost || – || 1-month black || – |- | Second wife for parent of a first wife || – || – || 3 months black || – |} The complexity of these etiquette rules extends to specific mourning periods and attire for siblings, step-parents, aunts and uncles distinguished by blood and by marriage, nieces, nephews, first and second cousins, children, infants, and "connections" (who were entitled to ordinary mourning for a period of "1–3 weeks, depending on level of intimacy"). Men were expected to wear mourning black to a lesser extent than women, and for a shorter mourning period. After the mid-19th century, men would wear a black hatband and black suit, but for only half the prescribed period of mourning expected of women. Widowers were expected to mourn for a mere three months, whereas the proper mourning period expected for widows was up to four years.<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–9}}</ref> Women who mourned in black for longer periods were accorded great respect in public for their devotion to the departed, the most prominent example being Queen Victoria herself. It was not uncommon for a widow who did not remarry to wear half-mourning for the rest of her life, except when another death necessitated full mourning. For example, Alexandra, Princess of Wales wore half-mourning for the rest of her life after her eldest son Eddy died in 1894. Empress Elisabeth of Austria did the same, as did Empress Eugénie of France. They reverted to full mourning when appropriate, but they never wore less than half-mourning after their sons' deaths. Women with lesser financial means tried to keep up with the example being set by the middle and upper classes by dyeing their daily dress. Dyers made most of their income during the Victorian period by dyeing clothes black for mourning.<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|page=341}}</ref> == Technological advancement == The technological changes that affected the manufacture and consumption of clothing in the Victorian age included the following: * the mass production of fabrics — for example, "by the early 1850s there were thousands of steam-powered looms churning out millions of miles of fabric every year"  [62] * the invention of aniline dyes, which were much more vibrantly colored and resistant to fading than the natural dyes that had been used. — . Invented by chemist [[William Henry Perkin]] in 1856, the first aniline dye mauveine (or mauve) "wash[ed] the fashionable landscape in a haze of purple."<ref name=":22">{{Cite book|title=The Dress Diary: Secrets from a Victorian Woman's Wardrobe|last=Strasdin|first=Kate|publisher=Pegasus Books|year=2023|location=New York, New York}}</ref> (247) Other intense and, to the Victorians, intensely exciting colors followed, but the new synthetic additions to fabric sometimes included chemicals harmful to their wearers. For example, a "bright-magenta hue was achieved by adding arsenical-based chemicals to existing aniline dyes, brightening the already luminous shades – but these left residues themselves, along with a toxic labour trail in their wake."<ref name=":22" /> (255) Perhaps the most famous of these is arsenic green, used on fabrics, wallpapers, and trim: "The craze for artificial foliage to adorn the heads and dresses of women of fashion in the mid-nineteenth century had seen the proliferation of flower workshops, where young women in their hundreds laboured to produce the lifelike green leaves and blooms that would make a fetching headdress or would trail becomingly across the bodice of a gown. The lushness of the green was achieved by the application of a powder, a pigment that was created by mixing copper and the highly toxic chemical, arsenic trioxide. The physical effects of working with this poisonous compound were horrific. Contemporary medical drawings depict the green hue of the skin and dreadful open lesions on the hands of the maker, whilst the daily gradual ingestion of the powder by the flower girls was eventually fatal."<ref name=":22" /> (255) * the invention of a sewing machine that could be used in the home. Although sewing machines were already in use in the clothing industry, in 1858 Isaac Merritt Singer began to sell "lightweight domestic machines" for home sewing, radically increasing women's control over their own dress.<ref name=":24">{{Cite book|title=Victorian Fashions for Women|last=Kay|first=Fiona|last2=Storey|first2=Neil R.|publisher=Pen & Sword History|year=2022|isbn=978 1 39900 416 9|location=Yorkshire and Philadelphia|pages=}}</ref> (91 [of 298]) * the spread of journalism for women and fashion journalism * the Jacquard loom: "French inventor Joseph-Marie Jacquard’s loom attachment mechanized the weaving of figured fabrics by 1804."<ref name=":25" /> (454) Perhaps not at the same scale as these but as important in 1850s designs was a technology that turned iron into steel, which could then be drawn into fine wires.<ref name=":3">{{Cite book|title=The Culture of Fashion|last=Breward|first=Christopher|publisher=Manchester University Press|year=1995|pages=145–180}}</ref> Steel was refined to a malleable state so that thin blades could be curved into concentric circles (called hoops) and connected with wires to form the cage. Technological advancements not only influenced the economy but brought a major change in the fashion styles worn by men and women. As the Victorian era was based on the principles of gender, race and class.<ref>{{cite journal|last1=Graham|first1=P|title=The Victorian Era|url=https://archive.org/details/in.ernet.dli.2015.261548|journal=Digital Library of India}}</ref> Much advancement was in favor of the upper class as they were the ones who could afford the latest technology and change their fashion styles accordingly. In 1830s there was introduction of horse hair crinoline that became a symbol of status and wealth as only the upper-class women could wear it. In 1850s there were more fashion technological advancements hence 1850s could rightly be called a revolution in the Victorian fashion industry such as the innovation of artificial cage crinoline that gave women an artificial hourglass silhouette without layers of petticoats, which was lighter and more hygienic.<ref>{{cite book|last1=Shrimpton|first1=J|title=Victorian Fashion|publisher=Bloomsbury Shire Publications}}</ref> Synthetic dyes, such as [[mauveine]] (aniline purple), were introduced in 1856, adding bright colours to garments. In 1855's ''[[Haute couture]]'' was introduced as tailoring became more mainstream in years to follow.<ref>{{cite book|last1=Aspelund|first1=Karl|title=Fashioning Society|publisher=Fairchild Books}}</ref> Charles Frederick Worth, a prominent English designer, became popular amongst the upper class though its city of destiny always is Paris. Haute couture became popular at the same time that sewing machines were invented.<ref name="Haute Couture">{{cite book|last1=Martin|first1=Richard|last2=Koda|first2=Harold|title=Haute Couture|publisher=The Metropolitan Museum of Art}}</ref> Princess [[Eugénie de Montijo|Eugenie]] of France wore the Englishman dressmaker, Charles Frederick Worth's couture and he instantly became famous in France though he had just arrived in Paris a few years ago. In 1855, Queen Victoria and Prince Albert of Britain welcomed [[Napoleon III]] and Eugenie of France to a full state visit to England. Eugenie was considered a fashion icon in France. Queen Victoria, who had been the fashion icon for European high fashion, was inspired by Eugenie's style and the fashions she wore.{{Citation needed|date=October 2025}} Later, Queen Victoria also appointed Charles Frederick Worth as her dress maker and he became a prominent designer amongst the European upper class. Charles Frederick Worth is known as the father of the haute couture as later the concept of labels were also invented in the late 19th century as custom, made to fit tailoring became mainstream.<ref>{{cite book|last1=Saillard|first1=Olivier|last2=Zazzo|first2=Anne|title=Paris Haute Couture|publisher=Skira Flammarion}}</ref> By the 1860s, when made-to-fit tailoring was popular in Europe, crinolines were considered impractical. In the 1870s, women preferred more slimmer silhouettes, hence bodices grew longer and the polonaise, a skirt and bodice made together, was introduced. In 1870s the Cuirass Bodice, a piece of armour that covers the torso and functions like a corset, was invented. Towards the end of Victoria's reign, dresses were flared naturally as crinolines were rejected by middle-class women. Designers such as Charles Frederick Worth were also against them. All these inventions and changes in fashion led to women's liberation as tailored looks improved posture and were more practical.<ref name="Haute Couture"/> dressmakers, couturiers, modistes == Home decor == {{main|Victorian decorative arts}} Home decor started spare, veered into the elaborately draped and decorated style we today regard as Victorian, then embraced the retro-chic of [[William Morris]] as well as pseudo-[[Japonaiserie]]. == Myths and Oversimplifications == === Modesty === {{main|Victorian morality}} {{Original research|section|date=May 2008}} [[File:1868-skirt-lengths-girl-ages-Harpers-Bazar.gif|thumb|upright|"The proper length for little girls' skirts at various ages", from ''[[Harper's Bazaar]]'', showing a 1900 idea of how the hemline should descend towards the ankle as a girl got older]]Many myths and exaggerations about the period persist to the modern day. Examples include the idea of men's clothing is seen as formal and stiff, women's as elaborate and over-done; clothing covered the entire body, and even the glimpse of an ankle was scandalous. Critics contend that [[corset]]s constricted women's bodies and lives. Homes are described as gloomy, dark, cluttered with massive and over-ornate furniture and proliferating [[bric-a-brac]]. Myth has it that even piano legs were scandalous, and covered with tiny [[pantalette]]s. === Tight Lacing === Tight-lacing, which was not possible until the development of the grommet in 1828, was famously controversial in the Victorian age, generating many column inches of profitable newspaper copy, in part because it was (and still is) fetishistic and subversive in that adolescent girls used it as a means of rebellion and upper-working- or lower-middle-class shop girls saw it as a means of upward mobility.<ref name=":21">{{Cite book|title=Fashion and Fetishism: Corsets, Tight-Lacing and Other Forms of Body-sculpture|last=Kunzle|first=David|publisher=History Press|year=2013|isbn=978 0 7524 9545 3|location=Stroud, Gloucestershire|pages=}}</ref> (71 [of 1182]) No evidence exists that tight lacing was widespread or particularly dangerous.<ref name=":21" /> () In truth, men's formal clothing may have been less colourful than it was in the previous century, but brilliant [[waistcoat]]s and [[cummerbund]]s provided a touch of colour, and [[smoking jacket]]s and [[robe|dressing gown]]s were often of rich Oriental [[brocade]]s. This phenomenon was the result of the growing textile manufacturing sector, developing mass production processes, and increasing attempts to market fashion to men.<ref name="shannon597"/> Corsets stressed a woman's sexuality, exaggerating hips and bust by contrast with a tiny waist. Women's [[evening gown]]s bared the shoulders and the tops of the breasts. The [[jersey dress]]es of the 1880s may have covered the body, but the stretchy novel fabric fit the body like a glove.<ref>{{cite book |last=Gernsheim |first=Alison |title=Victorian & Edwardian Fashion: A Photographic Survey |year=1981 |publisher=Dover Publications |location=New York |page=65|edition=New |isbn=0-486-24205-6}}</ref> Home furnishing was not necessarily ornate or overstuffed. However, those who could afford lavish draperies and expensive ornaments, and wanted to display their wealth, would often do so. Since the Victorian era was one of increased social mobility, there were ever more ''[[nouveaux riches]]'' making a rich show. The items used in decoration may also have been darker and heavier than those used today, simply as a matter of practicality. London was noisy and its air was full of [[soot]] from countless coal fires. Hence those who could afford it draped their windows in heavy, sound-muffling curtains, and chose colours that didn't show soot quickly. When all washing was done by hand, curtains were not washed as frequently as they might be today. There is no actual evidence that piano legs were considered scandalous. Pianos and tables were often draped with [[shawl]]s or cloths—but if the shawls hid anything, it was the cheapness of the furniture. There are references to lower-middle-class families covering up their [[pine]] tables rather than show that they couldn't afford [[mahogany]]. The piano leg story seems to have originated in the 1839 book, ''A Diary in America'' written by Captain [[Frederick Marryat]], as a satirical comment on American prissiness.<ref>{{cite book |last1=Marryat |first1=C.B. |title=A Diary in America: With Remarks on Its Institutions |date=1839 |publisher=Longman, Orme, Brown, Green, and Longmans |location=London, England |volume=2 |pages=246–247 |url=https://books.google.com/books?id=2-VEAAAAIAAJ&pg=PA246}} From pp. 246-247: "I was requested by a lady to escort her to a seminary for young ladies, and on being ushered into the reception-room, conceive my astonishment at beholding a square piano-forte with four ''limbs''. However, that the ladies who visited their daughters, might feel in its full force the extreme delicacy of the mistress of the establishment, and her care to preserve in their utmost purity the ideas of the young ladies under her charge, she had dressed all these four limbs in modest little trousers, with frills at the bottom of them!"</ref> Victorian manners may have been as strict as imagined—on the surface. One simply did not speak publicly about sex, childbirth, and such matters, at least in the respectable middle and upper classes. However, as is well known, discretion covered a multitude of sins. Prostitution flourished. Upper-class men and women indulged in [[adultery|adulterous]] liaisons. == Gallery == {{gallery |2=A mid-Victorian interior: ''Hide and Seek'' by [[James Tissot]], c. 1877 Image:Winterhalter Elisabeth.jpg|3=Dress designed by [[Charles Frederick Worth]] for [[Elisabeth of Bavaria|Elisabeth of Austria]] painted by [[Franz Xaver Winterhalter]].|4=File:Frith A Private View detail.jpg|5=[[William Powell Frith]]'s painting of 1883 contrasts women's [[Aesthetic dress]] (left and right) with fashionable attire (center).|6=File:Tissot lilacs 1875.jpg|7=Day dress, c. 1875 [[James Tissot]] painting.|8=File:James Abbot McNeill Whistler 011.jpg|9=[[James McNeill Whistler|Whistler]]'s [[Portrait of Lady Meux]], 1882 Image:Jeanna_Samary-Renoir.png|10=[[Pierre-Auguste Renoir|Renoir]]'s portrait of [[Jeanne Samary]] in an [[evening gown]], 1878|11=File:Melville_-_Queen_Victoria.jpg|12=Portrait by [[Alexander Melville (artist)|Alexander Melville]] of [[Victoria of the United Kingdom|Queen Victoria]], 1845|13=File:Henry Treffry Dunn Rossetti and Dunton at 16 Cheyne Walk.jpg|14=An artistic interior: [[Dante Gabriel Rossetti]] reading to [[Theodore Watts-Dunton]] in the drawing room at No. 16 [[Cheyne Walk]], 1882|15=File:Punch - Masculine beauty retouched1.png|16=Men's swimwear: Cartoon from ''[[Punch (magazine)|Punch]]'' by [[George du Maurier]]}} == See also == * [[Emily Clapham]] * [[Victorian decorative arts]] * [[Victorian dress reform]] * [[Victorian morality]] * [[Victoriana]] * [[Women in the Victorian Era]] * [[Charles Frederick Worth]] === Time periods === * [[1830s in fashion]] * [[1840s in fashion]] * [[1850s in fashion]] * [[1860s in fashion]] * [[1870s in fashion]] * [[1880s in fashion]] * [[1890s in fashion]] === Women's clothing === * [[Corset]] * [[Corset controversy]] * [[Tightlacing]] * [[Bloomers (clothing)|Bloomers]] * [[Bodice]] === Contemporary interpretations === * [[Steampunk]] * [[Neo-Victorian]] * [[Lolita Fashion|Lolita]] == References == {{Reflist}} == Further reading == *{{cite book |author=Phipps, Elena| title= ''From Queen to Empress: Victorian dress 1837-1877'' | location=New York | publisher=The Metropolitan Museum of Art | year=1988 | isbn=0870995340| url= http://libmma.contentdm.oclc.org/cdm/compoundobject/collection/p15324coll10/id/69547/rec/235 | display-authors=etal}} * Sweet, Matthew – ''Inventing the Victorians'', St. Martin's Press, 2001 {{ISBN|0-312-28326-1}} == External links == * [http://www.victorians.co.uk/victorian-fashion Victorian Fashion] {{Webarchive|url=https://web.archive.org/web/20180407223711/http://www.victorians.co.uk/victorian-fashion |date=7 April 2018 }} * [https://www.victorianvoices.net/topics/fashion/index.shtml VictorianVoices.net] – Fashion articles and illustrations from Victorian periodicals; extensive fashion image gallery * [http://www.cracked.com/article_19575_5-ridiculous-sex-myths-from-history-you-probably-believe.html Victorian myths] * [http://www.victorianstation.com/lifestylemenu.htm Victorian fashion, etiquette, and sports] {{Webarchive|url=https://web.archive.org/web/20180103162620/http://www.victorianstation.com/lifestylemenu.htm |date=3 January 2018 }} * [http://www.thesmartset.com/article/article12180701.aspx Background on "A Diary in America"] * [http://www.mccord-museum.qc.ca/en/keys/webtours/VQ_P2_17_EN.html Form and Fashion] — the evolution of women's dress during the 19th century (many photographs) * [http://www.mccord-museum.qc.ca/en/keys/games/jeu2/ Educational Game: Mix and Match] — build a 19th-century dress using a virtual mannequin * {{cite web |publisher= [[Victoria and Albert Museum]] |url= http://www.vam.ac.uk/content/articles/v/victorian-dress-at-v-and-a/ |title= Victorian Dress |work= Fashion, Jewellery & Accessories |date= 14 January 2011 |access-date= 2011-04-03}} *[http://cv.vic.gov.au/stories/creative-life/fashion-detective-fashion-fiction-and-forensics/ Fashion detective: Fashion, Fiction and Forensics in nineteenth century Australian fashion] on Culture Victoria {{Timeline of clothing and fashion|state=collapsed}}{{Victorian era|state=collapsed}} [[Category:Victorian fashion| ]] [[Category:19th-century fashion|*]] [[Category:1900s fashion]] [[Category:History of Western fashion]] [[Category:19th century in the arts]] =From ''Women in the Victorian era''= ===Victorian women's fashion=== {{Multiple issues|{{tone|date=March 2023}} {{more footnotes needed|date=March 2023}}|section=y}}{{Further|Victorian fashion}} The ideal Victorian woman was pure, chaste, refined, and modest. This ideal was supported by etiquette and manners. The etiquette extended to the pretension of never acknowledging the use of undergarments (sometimes generically referred to as "unmentionables"). The discussion of such a topic, it was feared, would gravitate towards unhealthy attention on anatomical details. As one Victorian lady expressed it: "[those] are not things, my dear, that we speak of; indeed, we try not even to think of them", in contrast to current norms.<ref>{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=20}}</ref> The pretence of avoiding acknowledgement of anatomical realities met with embarrassing failure on occasion. In 1859, the Hon. Eleanor Stanley wrote about an incident where the [[Louisa Cavendish, Duchess of Devonshire|Duchess of Manchester]] moved too quickly while manoeuvring over a [[stile]], tripping over her large [[hoop skirt]]: {{blockquote|[the Duchess] caught a hoop of her cage in it and went regularly head over heels lighting on her feet with her cage and whole petticoats above, above her head. They say there was never such a thing seen – and the other ladies hardly knew whether to be thankful or not that a part of her undergarments consisted in a pair of scarlet tartan [[knickerbockers (clothing)|knickerbockers]] (the things Charlie shoots in) which were revealed to the view of all the world in general and the [[Aimable Pélissier|Duc de Malakoff]] in particular".<ref>{{cite book|last=Cunnington|first=C. Willett|title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations|year=1990|publisher=Dover Publications|isbn=978-0-486-26323-6|pages=20–1}}</ref>}} However, despite the fact that Victorians considered the mention of women's undergarments in mixed company unacceptable, men's entertainment made great comedic material out of the topic of ladies' [[bloomers (clothing)|bloomers]], including men's magazines and music hall skits.<ref>{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=22}}</ref> Victorian women's clothing followed trends that emphasised elaborate dresses, skirts with wide volume created by the use of layered material such as [[crinoline]]s, hoop skirt frames, and heavy fabrics. Because of the impracticality and health impact of the era's fashions, a [[Victorian dress reform|dress reform movement]] began among women. The ideal silhouette of the time demanded a narrow waist, which was accomplished by constricting the abdomen with a laced [[corset]]. While the silhouette was striking, and the dresses themselves were often exquisitely detailed creations, the fashions were cumbersome. At best, they restricted women's movements and at worst, they had a harmful effect on women's health. Physicians turned their attention to the use of corsets and determined that they caused several medical problems: compression of the thorax, restricted breathing, organ displacement, poor circulation, and prolapsed uterus.<ref name="O'Connor"/> Articles advocating the reform of women's clothing by the British National Health Society, the Ladies' Dress Association, and the [[Rational Dress Society]] were reprinted in ''The Canada Lancet'', Canada's medical journal. In 1884, Dr J. Algernon Temple of Toronto even voiced concern that the fashions were having a negative impact on the health of young women from the working classes. He pointed out that a young working-class woman was likely to spend a large part of her earnings on fine hats and shawls, while "her feet are improperly protected, and she wears no flannel petticoat or woollen stockings".<ref name="O'Connor"/> [[File:Bloomers.jpg|thumb|1850s illustration of a woman wearing [[bloomers]]]] [[Florence Pomeroy]], Lady Haberton, was president of the Rational Dress movement in Britain. At a National Health Society exhibition held in 1882, Viscountess Haliburton presented her invention of a "[[divided skirt]]", which was a long skirt that cleared the ground, with separate halves at the bottom made with material attached to the bottom of the skirt. She hoped that her invention would become popular by supporting women's freedom of physical movement, but the British public was not impressed by the invention, perhaps because of the negative "unwomanly" association of the style with the American [[Bloomers]] movement.<ref>{{cite book|last=Murray|first=Janet Horowitz|title=Strong-Minded Women and Other Lost Voices from 19th Century England|year=1982|publisher=Pantheon Books|location=New York|isbn=0-394-71044-4|pages=[https://archive.org/details/strongmindedwome00jane/page/68 68–70]|url=https://archive.org/details/strongmindedwome00jane/page/68}}</ref> [[Amelia Jenks Bloomer]] had encouraged the wearing of visible bloomers by feminists to assert their right to wear comfortable and practical clothing, but it was no more than a passing fashion itself among radical feminists. The movement to reform women's dress would persist and have long-term success, however; by the 1920s, [[Coco Chanel]] was successful at selling a progressive, far less restrictive silhouette that abandoned the corset and raised hemlines. The new silhouette symbolised modernism for trendy young women and became the 20th century standard. Other Paris designers continued reintroducing pants for women and the trend was gradually adopted over the next century. Fashion trends, in one sense, travelled "full circle" over the course of the Victorian era. The popular women's styles during the [[Georgian era]], and at the very beginning of Victoria's reign, emphasized a simple style influenced by flowing gowns worn by women in [[Ancient Greek clothing|Ancient Greece]] and [[Clothing in ancient Rome|Rome]]. The [[Empire waist]] silhouette was replaced by a trend towards ornate styles and an artificial silhouette, with the restrictiveness of women's clothing reaching its low point during the mid-century passion for narrow corseted waists and hoop skirts. The iconic wide-brimmed women's hats of the later Victorian era also followed the trend towards ostentatious display. Hats began the Victorian era as simple [[Bonnet (headgear)|bonnets]]. By the 1880s, milliners were tested by the competition among women to top their outfits with the most creative (and extravagant) hats, designed with expensive materials such as silk flowers and exotic plumes such as ostrich and peacock. As the Victorian era drew to a close, however, fashions were showing indications of a popular backlash against excessive styles. Model, actress and socialite [[Lillie Langtry]] took London by storm in the 1870s, attracting notice for wearing simple black dresses to social events. Combined with her natural beauty, the style appeared dramatic. Fashions followed her example (as well as Queen Victoria's wearing of mourning black later in her reign). According to [[Harold Koda]], the former Curator-in-chief of the [[Costume Institute at The Met|Metropolitan Museum of Art's Costume Institute]],<ref>{{cite web|url=http://www.metmuseum.org/about-the-museum/press-room/exhibitions/2014/death-becomes-her|title=Death Becomes Her: A Century of Mourning Attire : October 21, 2014-February 1, 2015|website=Metmuseuim.org|access-date=7 November 2021}}</ref> "The predominantly black palette of [[mourning]] dramatizes the evolution of period silhouettes and the increasing absorption of fashion ideals into this most codified of etiquettes," said Koda, "The veiled widow could elicit sympathy as well as predatory male advances. As a woman of sexual experience without marital constraints, she was often imagined as a potential threat to the social order." ====Evolution of Victorian women's fashion==== <gallery> File:Fashion plate December 1844.jpg|Ladies' December Fashions (1844). Hand-coloured steel engraving from a women's magazine. File:Thegalleryofhmscalcutta james tissot 1876.jpg|''[[The Gallery of HMS Calcutta]]'' by [[James Tissot]] (1876). [[Bustle]]s were fashionable in the 1870s and 1880s. File:Mrs lillie langtry george frederic watts 1880.jpg|''Mrs. Lillie Langtry'' by [[George Frederic Watts]] (1880). File:Five-women-on-queenslander-steps-r.jpg|Fashionable women in [[Queensland]], Australia around 1900. </gallery> {{Short description|Irish writer (born 1963)}} {{Use Irish English|date=August 2025}} {{Use dmy dates|date=August 2025}} {{Infobox writer | name = Darach Ó Scolaí | image = Darach Ó Scolaí.JPG | alt = Man holding prize-winning book | caption = Ó Scolaí in 2019 | birth_name = Darach Ó Scolaí | birth_date = {{Birth date and age|1963|df=y}} | birth_place = [[County Galway]], The Republic of Ireland | death_date = | death_place = | occupation = Writer, artist, publisher | alma_mater = [[University of Galway]] | years_active = 1998–present | genre = Novel, retelling, translation, play, screenplay, illustrated book for children and adults | other_names = | spouse = | children = 3 | awards = [[Awards and Honors received by Darach Ó Scolaí|Awards and Honors]] | signature = | website = }}[[File:Darach Ó Scolaí.JPG|thumb|Darach Ó Scolaí, holding ''Oileán an Órchiste'' (his translation of Robert Louis Stevenson's ''Treasure Island'')]] == Darach Ó Scolaí == Darach Ó Scolaí (<small>Irish:</small> [/ˈda.rax/ /oː/ /sˠkˠoː/l̪ˠəi/]; born 1963<ref>{{Cite web|url=https://portraidi.ie/en/darach-o-scolai/|title=Darach Ó Scolaí|date=20 October 2017|website=Portráidí (Portraits of Irish-Language Writers)|access-date=1 August 2025}}</ref>) is an Irish author who works in a number of genres, from novels, plays and screenplays to illustrated books for children and adults. He began his literary career in 1998 writing screenplays, stage plays, retellings and translations; he began to publish novels in 2008. Ó Scolaí is widely recognized as a leading figure in contemporary Irish literature, known as “one of the most important Irish language writers of his generation”<ref>{{Cite journal|last=Poirtéir|first=Cathal|date=2022|title=? Suil an Daill: Constant Tensions and Shifting Allegiances|url=https://booksirelandmagazine.com/suil-an-daill-constant-tensions-and-shifting-allegiances/|journal=Books Ireland}}</ref> and "one of the great Irish language novelists [duine d’úrscéalaithe móra na Gaeilge]."<ref name=":17" /> His writing has been called “the high literature of the Irish language.”<ref>Ó Coimín, Maitiú. ''Nós'' 2 February 2018). Qtd. in "Táin Bó Cuailnge." ''Leabhar Breac''. Retrieved 25 August 2025.</ref> Much of his fiction is based on a knowledge of traditional Irish tales and narrative practices as well as Irish history. He specializes in literary and [[wikipedia:Historical_fiction|historical fiction]], or as novelist Alan Titley says, Ó Scolaí’s “peak (for now), or at least his greatest imaginative interest, is the historical novel [tá an chuma air gurb é a bhuaic (go fóill), nó ar a laghad, a mhórspéis samhlaíochta, an t-úrscéal staire].”<ref name=":11">{{Cite journal|last=Titley|first=Alan|date=Fall 2020|title=An Stíl Go Deo!: Soather Dharach Uí Scolaí (The style would be forever!: Worker Darach Ó Scolaí)|url=https://www.jstor.org/stable/27046090|journal=Comhar|volume=80, No. 10|pages=27|via=JSTOR}}</ref> His retellings of old stories and tales from their original Middle and Early-Modern Irish into Modern Irish ([[wikipedia:Irish_language|Gaeilge]]) are respected for their accessibility to students and language learners as well as for their artistry. Ó Scolaí also regularly reviews books and lectures and writes on literature and culture. Beyond his writing, Ó Scolaí is a publisher and has co-produced a number of film, television shows and stage plays. == Life == Ó Scolaí was born in Dublin and raised in the Galway [[wikipedia:Gaeltacht#Galway Gaeltacht|Gaeltacht]] (Irish-speaking) regions of Cois Fharraige on the north shore of Galway Bay, in the Republic of Ireland, where he lives now with his wife and children in Lochán Beag (Indreabhán).<ref name=":7">{{Cite journal|date=30 October 2024|title=Duais don úrscéal liteartha is fearr buaite ag Darach Ó Scolaí ag Oireachtas na Samhna|url=https://tuairisc.ie/duais-don-ursceal-liteartha-is-fearr-buaite-ag-darach-o-scolai-ag-oireachtas-na-samhna/|journal=Tuairisc}}</ref><ref>{{Cite journal|last=Ní Scolaí|first=Aifric|date=2024|title=Darach Ó Scolaí|url=https://www.taiscecf.ie/ealaiontoiri?category=Scr%C3%ADbhneoir|journal=Taisce Chois Fharraige}}</ref> He graduated the [[wikipedia:University_of_Galway|University of Galway]] (then University College Galway) with a B.A. in 1983.<ref>{{Cite web|url=https://www.linkedin.com/in/darach-ó-scolaí-20026920/|title=Darach Ó Scolaí|last=Ó Scolaí|first=Darach|date=August 2025|website=LinkedIn}}</ref> === Writing and Publishing === Ó Scolaí writes in Irish ([[wikipedia:Irish_language|Gaeilge]]), his native language, and lives in an area defined for the predominant presence of Irish as the vernacular language, the language spoken at home. Irish was the language of his parents' home and is the language of children as well. He is fluent in Irish and English and conversant in French. None of his works has been translated into English. ==== Leabhar Breac ==== In 1995 Darach Ó Scolaí and his brother Caomhán Ó Scolaí — a [[wikipedia:Typography|typographer]] and designer — founded the publishing house Leabhar Breac at Indreabhán (Inverin), County Galway. Their father “Séamas Ó Scolaí was an editor at An Gúm and worked on the Irish-English dictionary team [bhí a n-athair Séamas Ó Scolaí ina eagarthóir sa Ghúm agus d’oibrigh sé ar fhoireann an fhoclóra Gaeilge-Béarla].”<ref name=":0">{{Cite web|url=https://leabharbreac.com/en/about-us/|title=About Us|date=2024|website=Leabhar Breac|access-date=1 July 2025}}</ref> Darach Ó Scolaí has been publisher and literary editor at Leabhar Breac since its founding. Named for [[wikipedia:An_Leabhar_Breac|An Leabhar Breac (The Speckled Book)]], Leabhar Breac publishing house has more than 140 books in print.<ref name=":0" /> Leabhar Breac aims to publish Irish-language books that meet “a high literary and artistic standard.”<ref name=":0" /> Besides the content, Leabhar Breac is known for the typically "superb [thar cionn]" quality of the design and production of the "physical book [leabhar fisiciúil]."<ref name=":8">{{Cite journal|last=Ní Mhuilneoir|first=Gráinne|date=30 July 2024|title=‘Bláthnaid’ – leabhar álainn i sraithín álainn faoi mhná|url=https://tuairisc.ie/blathnaid-leabhar-alainn-i-sraithin-alainn-faoi-mhna/|journal=Tuairisc}}</ref> Its books regularly win awards for literary and artistic quality. Leabhar Breac also publishes translations for children and adults from various early versions of Irish as well as from French and English (and has published translations of books for young readers from Spanish, Catalan, and Italian as well). Leabhar Breac prints its books in Ireland. === Stage and Screen === ==== Rosg ==== In 1998 along with Ciarán Ó Cofaigh,<ref name=":1">{{Cite web|url=http://www.rosg.ie/en/about/History_6/|title=About Us: History|date=July 2025|website=Rosg|access-date=1 August 2025}}</ref> Ó Scolaí co-founded the film and television production company [http://www.rosg.ie/en/ Rosg] and was co-director until 2006. Rosg produced Ó ScolaÍ’s films ''Cosa Nite'' (1999), ''An Leabhar'' (2001) and ''Na Cloigne'' (2010). He left Rosg in 2006 to devote his time to other artistic activities. ==== Ealaín ar Oileán ==== In 2004, along with Val Balance, Ó Scolaí co-founded the annual artists' symposium Ealaín ar Oileán (trans., Art on an Island). The Irish-language symposium was held annually in the Áras Éanna arts and cultural center on Inis Oírr ([[wikipedia:Inisheer|Inisheer]], the smallest of the [[wikipedia:Aran_Islands|Aran Islands]]) from 2004 to 2013. Ó Scolaí was its co-director from its founding<ref>{{Cite web|url=https://ga.wikipedia.org/wiki/Darach_Ó_Scolaí.|title=Darach Ó Scolaí|date=3 February 2024|website=Vicipéid|access-date=1 July 2025}}</ref> until 2013. Besides being its co-director, Ó Scolaí has taken part in this conference as an artist<ref>{{Cite journal|date=16 January 2005|title=Darach Ó Scolaí|url=https://web.archive.org/web/20050116163252/http://bliainiris.com/authors/darach_oscolai.html|journal=Bliainiris}}</ref> and writer<ref name=":2">{{Cite web|url=http://ealainaroilean.ie/ealainaroilean.html|title=The Conference|date=7 September 2013|website=Ealaín ar Oileán|archive-url=https://web.archive.org/web/20130907083744/http://ealainaroilean.ie/ealainaroilean.html|archive-date=7 September 2013|access-date=1 August 2025}}</ref>. ==== Salamandar ==== In 2006 Ó Scolaí founded the stage production company Salamandar and directed his own play ''An Braon Aníos''. His plays ''An tSeanbhróg'' (2009) and ''Craos'' (2008) were also produced by Salamandar.<ref name=":19">{{Cite web|url=https://leabharbreac.com/en/product-category/darach-o-scolai/|title=Darach Ó Scolaí|date=2024|website=Leabhar Breac|access-date=1 July 2025}}</ref> == Works == === Novels === * [[wikipedia:An_Cléireach|''An Cléireach'' (trans., ''The Clerk'')]], Leabhar Breac, 2007. The Oireachtas Prize for Literary Fiction, 2007; The Ó Súilleabháin Award (Book of the Year) in 2008, and "named as ‘the best novel since the turn of the Century’ by Comhar."<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-cleireach/|title=An Cléireach - Leabhar Breac - Irish language novel|website=Leabhar Breac|language=en-US|access-date=2025-10-24}}</ref> * ''Na Comharthaí'' (trans., ''The Signs''), Leabhar Breac, 2014. * ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. The Oireachtas Prize for Literary Fiction, 2019.<ref name=":4">{{Cite web|url=https://leabharbreac.com/en/shop/fiction/suil-an-daill/|title=Súil an Daill|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Bódléar'', Leabhar Breac, 2024. The Oireachtas Prize for Literary Fiction, 2024<ref name=":7" />; The Ó Súilleabháin Award (Book of the Year) in 2025; featured in the 2025 Listen-Up Irish Summer Challenge for students of the Irish language.<ref>{{Cite news|url=https://connachttribune.ie/novel-approach-helps-people-learn-irish-in-a-creative-way/|title=Novel approach helps people learn Irish in a creative way|last=Murphy|first=Judy|date=3 October 2025|work=Connaught Tribune|access-date=24 October 2025}}</ref> === Retellings, Translations and Editions === The retellings and translations are into modern Irish. * ''Feis Tigh Chonáin'' (trans., ''The Feast of Conán's House''), Leabhar Breac, 2000; a retelling of a 15^th-century tale from the [[wikipedia:Fenian_Cycle|Fenian Cycle]].<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/feis-tigh-chonain/|title=Feis Tigh Chonáin|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''An Ceithearnach Caolriabhach'' (trans., ''The Narrow-Striped Kern''), Leabhar Breac, 2002; a retelling from c. 1500, also illustrated by Darach Ó ScolaÍ.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-ceithearnach-caolriabhach/|title=An Ceithearnach Caolriabhach|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), Leabhar Breac, 2017, both a modern edition of an 11th-century epic and an annotated edition.<ref name=":3">{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/tain-bo-cuailnge-2-2/|title=Táin Bó Cuailnge|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> "''Táin Bó Cuailnge'' won the Aodán Mac Poilín Memorial Prize 2017."<ref name=":19" /> * ''Deirdre'', Leabhar Breac, 2023, a “picture book for adults” with artist Anastasia Melnykova.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/deirdre/|title=Deirdre|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Part of the [[wikipedia:Ulster_Cycle|Ulster Cycle]], ''Deirdre'' is a retelling of the story of possibly the most widely known Irish figure from the early tales and sagas.<ref>{{Cite book|title=A Dictionary of Celtic Mythology|last=MacKillop|first=James|publisher=Oxford University Press|year=2004|isbn=9780198609674|pages=181}}</ref> * ''Bláthnaid'', Leabhar Breac, 2024, a “picture book for adults” with artist Anastasia Melnykova; “one of the great stories of the [[wikipedia:Ulster_Cycle|Ulster Cycle]].”<ref name=":4" /> * ''Sadhbh,'' Leabhar Breac, 2025, a picture book for adult readers, illustrated by Alé Mercado; a retelling of the medieval tale ''Ceasacht Inghine Ghuile (''trans., ''The Complaint of Guile's Daughter'').<ref name=":5">{{Cite web|url=https://leabharbreac.com/en/tales-of-wonder/|title=Tales of Wonder|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Eoghan Béal'', Leabhar Breac, 2025, a picture book for adult readers illustrated by Alé Mercado<ref name=":5" />; a retelling of the medieval tale ''[https://ga.wikipedia.org/wiki/Caithr%C3%A9im_Cellaig Cathréim Ceallaigh]'' from ''The Yellow Book of Leacan.''<ref name=":5" /> === For Young Readers === Ó Scolaí has written illustrated books for young readers (8–10 years old) in two series, the Fionn Series and the Scéalta Staire series, and translated a large number of classics and popular books for children of all ages. The number of these written and translated works suggests a commitment to children and their literacy in Irish. The Fionn Series “is a retelling ... of the great legends of the Fianna for the young Irish readers of today.”<ref name=":6">{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/doiteoir-na-samhna/|title=Dóiteoir na Samhna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> [[wikipedia:The_Boyhood_Deeds_of_Fionn|Macgnímartha Finn (The Boyhood Deeds of Fionn)]] is a medieval story in the [[wikipedia:Fenian_Cycle|Fenian Cycle]]. * ''An Bradán Feasa'' (trans., ''The Salmon of Knowledge''), Leabhar Breac, 2010, “shortlisted for the Réics Carlo award 2010.”<ref name=":9">{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/an-bradan-feasa/|title=An Bradán Feasa|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Dóiteoir na Samhna'' (trans., ''The Halloween Burner''), 2010.<ref name=":6" /> * ''Bodach an Chóta Lachna'' (trans., ''The Churl in the Dun Coat''), 2011.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/|title=Bodach an Chóta Lachna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> The Scéalta Staire (Historical Stories) series<ref name=":9" /> * ''Mánas Ó Dónaill'', 2000. * ''Seán Ó Néill'', Leabhar Breac, 2000. * ''Gráinne Mhaol Ní Mháille'', Leabhar Breac, 2003. * ''Tadhg Dall Ó hUiginn'', Leabhar Breac, 2003. ==== Translations ==== * Robert Louis Stevenson, ''Oileán an Órchiste'' (trans. of ''Treasure Island''), Leabhar Breac, 2014.<ref>{{Cite journal|date=2025-06-19|title=Oireachtas na Gaeilge|url=https://en.wikipedia.org/w/index.php?title=Oireachtas_na_Gaeilge&oldid=1296394643|journal=Wikipedia|language=en}}</ref> * Robert Louis Stevenson, ''An Fuadach'' (trans. of ''Kidnapped''), Leabhar Breac, 2016. * Clement Clarke Moore, ''Cuairt San Nioclás'' (trans. of ''A Visit from St. Nicholas'', or "'Twas the Night Before Christmas"), Leabhar Breac, 2022. '''''The Corto Maltese Graphic Novels''''' Written in Italian by Hugo Pratt and translated by Ó Scolaí, both adults and teenagers read this series of Italian adventure graphic novels.<ref>{{Cite journal|date=2025-07-01|title=Corto Maltese|url=https://en.wikipedia.org/w/index.php?title=Corto_Maltese&oldid=1298285365|journal=Wikipedia|language=en}}</ref> Ó Scolaí's '''translation of ''Corto Maltese''''' was listed in 2017 among "The 30 Irish books that Irish people love."<ref>{{Cite journal|last=Ó Murchú|first=Eoin P.|date=09/06/2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil [The 30 Irish books that Irish people love]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}</ref> * Hugo Pratt, ''Corto: Port na Farraige Goirt'', Leabhar Breac, 2013. * Hugo Pratt, ''Corto: The Golden House in Samarkand'', 2014. * Hugo Pratt, ''Corto: Na Liopard-Fhir ó Rufiji'' (trans. of ''Corto: The Leopard Men of Rufiji''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: In Ainm Dé Uilthrócairigh'' (trans. of ''Corto: In the Name of God All-Merciful''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: Tóraíocht Eile'' (trans. of ''Corto: Another Quest''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: Sa tSibéir'' (trans. of ''Corto: In Siberia''), Leabhar Breac, 2016. '''''Other Translations for Children''''' Ó Scolaí has translated into Irish six books from the ''Le Pavillon Noir'' (trans., ''Jolly Roger'') series by Alain Surget; four books from the ''Catalan First Steps'' series by Enric Lluch Girbés and the ''Caitlín & Cormac'' series by Joan Carles; three books from the ''Louisette le Taupe'' series by Bruno Heitz, and three books from the ''Loup'' series by Orianne Lallemand. === Plays and Screenplays === ==== Stage Plays ==== Ó Scolaí was writer and director of the original productions of two plays in the ''Trí Bhraon'' (trans., ''Three Drops'') trilogy; ''Coinneáil Orainn'' was directed by Darach Mac Con Iomaire and staged by An Taibhdhearc. All three plays have been published in book form by Leabhar Breac. * ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32553|title=Coinneáil Orainn|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}</ref> The first play in the ''Trí Bhraon'' (''Three Drops'') trilogy. [[wikipedia:Taibhdhearc_na_Gaillimhe|An Taibhdhearc]], the national Irish-language theatre of Ireland, toured the country in 2005 with ''Coinneáil Orainn''.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/coinneail-orainn/|title=Coinneáil Orainn|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Walter Macken Prize, 2005; BBC Stewart Parker Award, 2006.<ref>{{Cite web|url=https://irishplayography.com/person/darach-scola|title=Darach Ó Scolaí|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}</ref> * ''Branwen'', 2006, by Darach Ó Scolaí and Ifor ap Glyn, in Irish, Welsh and English, co-produced by Project Arts Centre and Llwyfan Gogledd Cymru, toured the Republic of Ireland and Wales.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32418|title=Branwen|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> * ''An Braon'' Aníos (trans., ''Rising Damp''), 2006, directed by Ó Scolaí.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32461|title=An Braon Aníos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> The second play in the ''Trí Bhraon'' (''Three Drops'') trilogy. “The Salamandar company toured the country in 2006-07 with this play, and Salamandar also produced a radio version of the play for RTÉ Raidió na Gaeltachta in 2009.”<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/an-braon-anios/|title=An Braon Aníos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Craos'' (trans., ''Gluttony''), 2008, directed by Ó Scolaí.<ref name=":13">{{Cite web|url=https://irishplayography.com/play?playid=32867|title=Craos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> The third play in the ''Trí Bhraon'' (''Three Drops'') trilogy, it toured to Cork and Belfast.<ref name=":13" /> A review of the 2008 Salamander performance in the ''Irish Times'' says, “a humorous play which offers plenty to think about, fine acting, and sparklingly witty dialogue.”<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/craos-2/|title=Craos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''A+E'', 2008, by Ríonach Ní Néill and Darach Ó Scolaí, "dance and music drama," co-produced by Ciotóg and Salamandar.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32962|title=A+E|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> * ''An tSeanbhróg'' (trans., ''The Old Shoe''), 2009, produced by Salamander<ref>{{Cite web|url=https://irishplayography.com/play?playid=33042|title=An tSeanbhróg|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> and staged in the Axis Arts Centre, Dublin, and the Letterkenny Arts Centre. * '''In ''Mhuir Fhíondorcha/The Wine-Dark Sea: The Homer Project'', Ó Scolaí's translation of Homer's Cyclops story, performed at the 2019 IMRAM festival'''.<ref>{{Cite news|url=https://www.irishtimes.com/culture/books/imram-a-festival-celebrating-the-irish-language-1.4047610|title=Imram: a festival celebrating the Irish language. Liam Carson reveals the myths and legends appearing in this year’s programme|last=Carson|first=Liam|date=11 October 2019|work=The Irish Times|access-date=15 October 2025}}</ref> ==== Screenplays ==== * ''Cosa Nite'' (trans., ''Washed Feet''), short film, 1998 (dir. Dearbhla Walsh, prod. Ciarán Ó Cofaigh, Rosg); "a prose version of ''Cosa Nite'' was published (Rosg 2000)."<ref name=":9" /> Nominated for an Irish Film and Television Award.<ref>{{Citation|title=Cosa Nite (Short 1998) - Awards - IMDb|url=https://www.imdb.com/title/tt0191917/awards/|accessdate=2025-08-25|language=en-US}}</ref> * ''Na Glúnta'' (trans., ''The Generations''), 2001<ref>{{Cite web|url=https://www.iftn.ie/production/production_companies/production_sub/feature/?act1=record&aid=70&rid=3917&tpl=filmography_dets&only=1&force=1|title=Na Glúnta {{!}} The Irish Film & Television Network|website=www.iftn.ie|access-date=2025-08-25}}</ref>, co-directors Ciarán Ó Cofaigh & Darach Ó Scolaí, prod. Ciarán Ó Cofaigh, Rosg. * ''An Leabhar'' (trans., ''The Book''), short film, 2000, (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg) Rosg, 2000.<ref>{{Citation|title=An Leabhar|url=https://www.imdb.com/title/tt0963767/|publisher=Bord Scannán na hÉireann / The Irish Film Board, ROSG|accessdate=2025-08-25|first=Robert|last=Quinn|others=Colm O&apos;Maonlai, Peadar O&apos;Treasaigh, Diarmuid Mac an Adhastair}}</ref> * ''Na Cloigne'' [trans., The Heads], 3-episide series, 2010 (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg), TG4.<ref>{{Cite web|url=https://www.imdb.com/title/tt1607924/|title=Na cloigne|date=2010|website=IMDb|access-date=25 August 2025}}</ref> === Nonfiction === Ó Scolaí's essays and lectures are published and his interviews are broadcast regularly, making for a large body of nonfiction critical and analytical work. Here are a few, almost all published in [https://comhar.ie/iris/scribhneoiri/darach-o-scolai/ Comhar]: * “Ceol Ciúin na nÉagmaise” (trans., “The Silent Music of Absence ['''the Fall?''']”), an essay on the 2014 Nobel Prize winner for literature, [[wikipedia:Patrick_Modiano|Patrick Modiano]], ''Comhar'', December 2014. * The Ó Cadhain Lecture: [https://leachtaiuichadhain.clo.ie/leachtai/2014 “Cuimhne agus Díchuimhne (trans., “Memory & Forgetfulness"]), 2014. * “Rithim agus Réim” ("Rhythm and Register"), a public lecture in the University College Dublin lecture series “Ó Thrácht go Twitter” (trans., "From Talk to Twitter"), 2014. * Review of Pádraig Ó Cíobháin’s ''Dréachta Chrích Fodla'', '''Comhar?, ??'''. * “Na Geilt i mBun an Tí” (trans., "The Madmen in Charge"), a talk at the Merriman Winter School, Comhar April 2012.<ref name=":18">{{Cite web|url=http://darachoscolai.ie/beathaisneis.html|title=Darach Ó Scolaí: Beathaisnéis|website=darachoscolai.ie|access-date=2025-09-26}}</ref> * The EFACIS podcast: Síle Ní Choincheannain talks to Darach Ó Scolaí about the historical novel. == Critical Reception == Ó Scolaí’s style has been called “crisp and elegant, and rich in language while being highly readable,”<ref>{{Cite journal|last=Heussaf|first=Anna|date=Summer 2025|title=Bláthnaid—A tale of love, violence and sorcery retold for readers today|url=https://booksirelandmagazine.com/blathnaid-a-tale-of-love-violence-and-sorcery/|journal=Books Ireland}}</ref> with “an unsurpassed richness and precision of language.”<ref name=":12">{{Cite journal|last=Ó Cróinín|first=Breandán|date=Summer 2025|title=unknown|journal=The Limerick Leader}}</ref> “Whimsical, hilarious, and subtly learned” is how Éilis Ní Dhuibhne described his writing.<ref name=":20" /> === Original Works === Ó Scolaí’s first novel, the 2007 ''An Cléireach'' (''The Clerk'') won two prizes and was described as “one of the great historical novels in the Irish language and among the best books written in the language since the beginning of this century.”<ref name=":12" /> Novelist Alan Titley says, “In ''An Cléireach'' Ó Scolaí creates the Ireland of war in the 17th century more fully than any other Irish writer on the subject of war since ''L’Attaque'' Eoghain Ó Thuairisc around 1798 [In ''An Cléireach'' cruthaíonn Ó Scolaí Éire an chogaidh san 17ú haois níos iomláine ná mar a dhein aon scríbhneoir Gaeilge eile ar ábhar cogaidh ó ''L’Attaque'' Eoghain Uí Thuairisc timpeall ar 1798].”<ref name=":11" />{{rp|25, Col. 1a}} Not all the reviews of this first novel were so positive, however; Proinsias O' Drisceoil says for the Irish Times says,<blockquote>This then is a novel in search of a plot, a story that attempts to attain a significance that eludes it.<ref>{{Cite news|url=https://www.irishtimes.com/news/a-disaffected-clerk-in-the-confederates-1.943070|title=A disaffected clerk in the confederates|last=O' Drisceoil|first=Proinsias|date=5 July 2008|work=The Irish Times|access-date=16 October 2025}}</ref></blockquote> In the ''Oxford Handbook of Modern Irish Fiction'' Pádraig Ó Siadhail analyzes rather than reviews ''An Cléireach'': <blockquote>In ''An Cléireach'', Ó Scolaí revisits the trauma of Cromwellian Ireland. The primary narrative device is once again the first-hand account, in this case by Tadhg Ó Dúbháin, a clerk and quartermaster in the Confederate Army in 1650. We sample the hardships, the friendships, the tensions, the rivalries, and the petty jealousies amongst comrades in arms, including remnants of the Gaelic literary class, as the Confederate soldiers, increasingly a rabble more than a cohesive unit, retreat in advance of Cromwell’s forces. ''An Cléireach'' concludes with the narrator and his family in exile in continental Europe. But along the retreat route, and central to the novel, members of the Confederate army camp, rest up, and tell versions of a story about the keeper of the treasured manuscript "Saltair an Easpaig" (The Bishop’s Psalter). Their versions raise issues about memory construction, the limitations of individual perspectives, personal agendas, and how minor changes in the telling of a story can alter our understanding of history, Thus, ''An Cléireach'' complements ''Fontenoy'' in moving beyond more realistic recreation of a historical event or period to interrogate the notion of history as construct.<ref>{{Cite book|title=The Oxford Handbook of Modern Irish Fiction|last=Ó Siadhail|first=Pádraig|publisher=Oxford University Press|year=2020|isbn=9780198754893|editor-last=Harte|editor-first=Liam|pages=598–99|chapter=Contemporary Irish Fiction}}</ref> </blockquote> Of ''Súil an Daill,'' in ''Nós'', Cathal Seoighe says, "The book deserves a significant place among the collection of high-quality books published in recent years that would make you feel sorry for someone who does not speak Irish [Tá áit shuntasach ag dul don leabhar i measc an chnuasaigh leabhair ar ardchaighdeán a foilsíodh le roinnt blianta anuas a d’fhágfadh trua agat don té atá gan Ghaeilge]."<ref>{{Cite journal|last=Seoighe|first=Cathal|date=09/26/2022|title=‘Dar leathmhagairle an diabhail, is leabhar den scoth é seo!’ ['According to the devil’s half-wit, this is a great book!’]|url=https://nos.ie/cultur/leabhair/dar-leathmhagairle-an-diabhail-is-leabhar-den-scoth-e-seo/|journal=Nós}}</ref> ''Bódléar'', Ó Scolaí's most recent book, is a “beautiful novel. There is magic and craftsmanship in it. A small miracle of a book and it is highly recommended.”<ref>{{Cite web|url=https://leabharbreac.com/bodlear-mioruilt-bheag-de-leabhar/|title=Bódléar: Míorúilt bheag de leabhar (Bódléar: A Small Miracle of a Book)|last=Ní Ghairbhí|first=Róisín|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Éilis Ní Dhuibhne in the ''Irish Times'' says,<blockquote>what a gem! An affectionately gentle satire of the Irish poetic scene during one creatively fluid 19th-century year, the story focuses on a Maigue poet and schoolteacher who goes on a trip to France and returns with camembert, a cafetiere, ‘Fleurs du Mal’, and a mission to convert the local traditionalists to la modernité. Whimsical, hilarious, and subtly learned, it’s absolutely delightful!<ref name=":20">{{Cite journal|last=Ní Dhuibhne|first=Éilis|date=30 June 2025|title=Éilís Ní Dhuibhne on the best Irish language books of 2025 so far: Including a history of the Gaeltacht Civil Rights Movements, a gem of a novel by Darach Ó Scolaí and Joe McHugh’s entertaining account of learning Irish|url=https://www.irishtimes.com/culture/books/review/2025/06/30/eilis-ni-dhuibhne-on-the-best-irish-language-books-of-2025-so-far/|journal=The Irish Times|pages=22}}</ref></blockquote> === Retellings and Translations === ==== ''Táin Bó Cuailnge'' ==== ''Táin Bó Cuailnge'' [''The Cattle Raid of Cooley''] is a modern edition of an 11th-century epic into modern Irish.<ref name=":3" /> Gearóid Denvir reviewed ''Táin Bó Cuailnge'' for ''Comhar'':<blockquote>Darach Ó Scolaí has ​​achieved a feat in this challenging reworking. He has found a high level of the Irish language to tell his story – as he has done before in his groundbreaking novel An Cléireach (2007, Leabhar Breac) and in his other prose works. This book is a decoration of the language, literature and culture of the Irish language, following the path of the old storytellers and writers and presenting material from the tradition to his own generation according to the understandings of his own time. The book will be a classic that will be of great interest to all readers of the Irish language, both ordinary readers, students, scholars and writers, and there should be a copy in every home in the country. [Tá éacht déanta ag Darach Ó Scolaí san athleagan dúshlánach seo. Tá réim ard den teanga Ghaeilge aimsithe aige lena scéal a inseacht – mar a rinne sé cheana ina úrscéal ceannródaíoch An Cléireach (2007, Leabhar Breac) agus i saothair eile phróis dá chuid. Is maisiú ar an teanga agus ar litríocht agus cultúr na Gaeilge an leabhar seo a leanas conair na seanscéalaithe agus na seanscríobhaithe agus ábhar de chuid an traidisiúin á chur i láthair a ghlúine féin aige de réir thuiscintí a linne féin. Clasaic a bheas sa leabhar a gcuirfidh léitheoirí uilig na Gaeilge, idir ghnáthléitheoirí, mhic léinn, scoláirí agus scríbhneoirí spéis thar na bearta ann, agus ba cheart cóip a bheith i chuile theach sa tír.]<ref name=":15">{{Cite journal|last=Denvir|first=Gearóid|date=April 2018|title=Táin Bó Cuailgne|url=https://comhar.ie/iris/78/4/leirmheas/|journal=Comhar|via=JSTOR}}</ref> </blockquote>Cathal Poirtéir says, "The freshness and richness of Ó Scolaí’s version are a joy …. The author delights us with the linguistic and stylistic richness of the ancient epic in a modern-Irish version that reflects the original’s spirit and language."<ref>{{Cite journal|last=Poirtéir|first=Cathal|date=May/June 2018|title=Leabhair Idir Lámha|url=https://www.jstor.org/stable/26564180|journal=Books Ireland|pages=46–47|via=JSTOR}}</ref>{{rp|47}} Novelist and academic Alan Titley calls Ó Scolaí's "a wonderful gutsy telling" of ''Táin Bó Cuailnge''.<ref>{{Cite news|url=https://www.irishtimes.com/culture/2023/03/11/the-tain-retold-maeve-and-ailills-spat-could-be-out-of-a-soap-opera/|title=The Táin retold: ‘Maeve and Ailill’s spat could be out of a soap opera’|last=Titley|first=Alan|date=11 March 2023|work=The Irish Times|access-date=16 October 2025}}</ref> ==== ''Deirdre'' ==== Marie Whelton, in "Léann Teanga" ("Language Studies"), in the 2024 ''An Reiviú'' says,<blockquote>this version [of ''Deirdre''] by Darach Ó Scolaí succeeds in skillfully capturing and portraying the complexity of gender and power issues in the ‘Deirdre’ tradition [éiríonn leis an leagan seo le Darach Ó Scolaí castacht cheisteanna na hinscne agus na cumhachta i dtraidisiún scéal Dheirdre a ghabháil agus a léiriú go sciliúil]. … There is no doubt that this new version greatly contributes to the legacy of the story and that it revives that legacy thoughtfully and artistically [Níl amhras faoi ach go gcuireann an leagan úr seo go mór le hoidhreacht an scéil agus go ndéanann sé an oidhreacht sin a athbheochan go tuisceanach agus go healaíonta.].<ref name=":16">{{Cite web|url=https://www.tara.tcd.ie/tara8/server/api/core/bitstreams/3c20175a-7631-44b2-8b0f-f454edd712b4/content|title=An Artistic Retelling of Deirdre's Tale and the Defeat of Conor Review of Deirdre or the Ship of Mac Uisnigh by Darach Ó Scolaí [Athinsint Ealaíonta ar Oidhe Dheirdre agus ar Ansmacht Chonchúir Léirmheas ar Deirdre nó Loingeas Mhac Uisnigh le Darach Ó Scolaí]|last=Whelton|first=Marie|date=2024|website=The Review [An Reiviú], Language Studies [Léann Teanga]|access-date=25 September 2025}}</ref></blockquote> === Works for Young Readers === Meadhbh Ní Eadhra said of ''Bodach an Chóta Lachna'' that it was "Beautiful Irish, but easy to understand for young readers."<ref>Ní Eadhra, Meadhbh. In ''Gaelscéal'', qtd. in "Bodach an Chóta Lachna" https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/.</ref> == Awards and Honors == Ó Scolaí's works are regularly nominated and make the short list for prizes, an honor in itself, but they are generally not listed here unless they are named as the first-place winner in their category. === Oireachtas Prize === The Oireachtas Prize is the literary prize awarded by [[wikipedia:Oireachtas_na_Gaeilge|Oireachtas na Gaeilge]], the annual arts festival dedicated to Irish language, arts and culture. Darach Ó Scolaí has won the Oireachtas Prize for Literary Fiction three times, once for ''An Cléireach'' (''The Clerk'') in 2007, for ''Súil an Daill'' (''The Eye of the Blind'') in 2021 and for ''Bódléar'' in 2024. * 2007, for ''An Cléireach'' (trans., ''The Clerk'') — “(a special prize commemorating the 400th anniversary of the foundation of Coláiste na nGael in Louvain, awarded under the auspices of the Franciscan Province of Ireland). The prize of €10,000 was the largest prize ever awarded to an Irish language novel [(duais speisialta chomórtha 400 bliain bhunú Choláiste na nGael i Lobháin a bronnadh faoi urraíocht Phroibhinse Phroinsiasach na hÉireann). Ba é an duais €10,000 sin an duais ba mhó a bronnadh riamh ar úrscéal Gaeilge].”<ref name=":18" /> * 2021, for ''Súil an Daill'' (''The Eye of the Blind'') * 2024, for ''Bódléar'' === Ó Shúilleabháin Award, Irish language “Book of the Year” === The first prize of this award includes €5,000 to the publisher and €2,500 to the author of the winning work.<ref name=":10">{{Cite journal|date=15 August 2023|title=20 saothar san iomaíocht do ‘Leabhair Ghaeilge na Bliana 2023’|url=https://tuairisc.ie/20-saothar-san-iomaiocht-do-leabhair-ghaeilge-na-bliana-2023/|journal=Tuairisc}}</ref> * ''An Cléireach'' (''The Clerk'').<ref>{{Cite web|url=http:/www.gaelport.com/uploads/documents/edition19.html|title=Eagrán / Edition 19 - 04 11 2008|date=4/11/2008|website=Internet Archive|archive-url=https://web.archive.org/web/20130525011340/http:/www.gaelport.com/uploads/documents/edition19.html|archive-date=25 May 2013|access-date=25 August 2025}}</ref> * ''Táin Bó Cuailnge'', 2018. * ''Bódléar'', 2025. ==== De Bhaldraithe Award ==== The Gradam de Bhaldraithe is awarded to the best work in translation.<ref name=":10" /> * ''Cuairt San Nioclás,'' a translation of Clement Clarke Moore's ''A Visit from St. Nicholas'', or "'Twas the Night Before Christmas."<ref name=":10" /> ==== Other ==== * Walter Macken Prize, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005 * Bháiteir Uí Mhaicín Memorial Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005<ref>{{Cite news|url=https://www.irishtimes.com/gaeilge/tuarascail/duais-oireachtais-1.501571|title=Oireachtas Prize: Over €50,000 was awarded to writers in the Oireachtas Literary Competitions at an event in Dublin last night. Winners… [Duais Oireachtais: Bronnadh breis agus €50,000 ar scríbhneoirí i gComórtais Liteartha an Oireachtais ar ócáid i mBaile Átha Cliath aréir. Bhuaigh…]|work=5 October 2005|access-date=15 October 2025}}</ref> * BBC Stewart Parker Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2006 * The Aodán Mac Póilín Commemorative Prize, for ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), 2017 == External Links == * Leabhar Breac website: https://leabharbreac.com/en/ * Leabhar Breac Facebook pages: * Rosg website: [http://www.rosg.ie/en/ <nowiki>http://ww</nowiki>w.rosg.ie/en/] * Art on the Island (Ealaín ar Oileán) website, archived at the Wayback Machine: https://web.archive.org/web/20130601000520/http://ealainaroilean.ie/ 31 March 2012, 1 June 2013 and 8 January 2014 * Darach Ó Scolaí's website Archived 25 September 2015 at the Wayback Machine: https://web.archive.org/web/20150925103456/http://darachoscolai.ie/ * Youtube video of [https://www.youtube.com/watch?v=OlP2AmSBzXc Breandán Ó Cróinin introducing Deirdre at the book launch] in the pub Tigh Mholly (Molly’s House). == Primordial Ooze == * Known for his sensitivity to language and voices. * Finish scanning through JSTOR * Scan through Irish Times, 56 hits * Check Goodreads * Check YouTube (In the spring of 2013, the arts programme Imeall interviewed the author on TG4.) * Check both Wikipedias for pages on the origins of the retold tales (like Deirdre) and link to this article * Propose link from University of Galway page once Darach’s is up * Write Irish National Biography (<nowiki>https://www.dib.ie</nowiki>) to propose an article about Darach once the Wikip article is done? See what they say. * Link to Ó Scolaí from the Wikipedia * Make sure links '''to''' Wikipedia in the actual encyclopedia work right === Not Placed Yet === * "So here are the books that Irish people love the most! [Mar sin seo iad na leabhair is gile leis na Gaeil!]" — "32. An Cléireach – Darach Ó Scolaí (2)" [18 books got 2 votes, and then they're alphabetized by author's last name, so the 32 of 34 doesn't signify the specificity it seems to]<ref name=":14">{{Cite journal|last=Ó Murchú|first=Eoin P.|date=9 June 2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil. [The 30 best Irish books for Irish people]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}</ref> * "Below is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers.Here is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers [Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí.Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí]." "Corto Maltese – Hugo Pratt (aistrithe ag Darach Ó Scolaí)"<ref name=":14" /> * "Ceann eile de bhuaicphointí na hÉigse a bheidh sa seisiún le Darach Ó Scolaí, duine d’úrscéalaithe móra na Gaeilge, agus duine de chomhbhunaitheoirí teach foilsitheoireachta Leabhar Breac. [Another highlight of the Éigse will be the session with Darach Ó Scolaí, one of the great Irish language novelists, and one of the co-founders of the publishing house Leabhar Breac.]"<ref name=":17">{{Cite journal|last=Nós|date=4 May 2023|title=Éigse na Bruiséile le filleadh i mí na Bealtaine. [Éigse na Bruséile to return in May]|url=https://nos.ie/cultur/eigse-na-bruiseile-le-filleadh-i-mi-na-bealtaine/|journal=Nós}}</ref> === Things Taken Out for Now === “’The play is a comedy about language, lies, bureaucracy and Gaeltacht grants, in the tradition of Myles na Gcopaleen,’ according to Norma-Jean Kenny in the ''Galway Advertizer'', ‘in which the author comments and criticizes the institutions of the Irish language in Ireland without ceasing.’" Supposedly a quotation by Gearóid Denvir reviewing ''Táin Bó Cuailnge'' for ''Comhar'' (but I don't find it in the article): This book has long been needed by Irish language readers and there is no doubt that it will become a classic in time and surpass Thomas Kinsella’s English version. This version remains faithful to the language of the original while at the same time finding an appropriate language in today’s Irish. Ó Scolaí masterfully overcomes the difficulties of the original’s rhetorical difficulties and the versions of the original poetic texts are extremely effective.[supposedly <ref name=":15" />] “The biggest prize ever awarded for a novel in Irish was presented at a special ceremony in the National Concert Hall in Dublin, today (Thursday, 4 October 2007). Darach Ó Scolaí, writer, artist & playwright from Casla, Co. Galway, was awarded €10,000 for his literary novel, ‘An Ardscoil’. This work, under the new title ‘An Cléireach’, will be launched at Oireachtas na Samhna in Westport in November. This is the first novel from his pen, a story set in the late seventeenth century. This competition was sponsored by the Franciscan Province of Ireland.” (archive, Oireachtas na Gaeilge site, 04 October, 2007) ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. number 2 in ''Comhar'' literary magazine’s list of best books of 2021. '''{6}.''' *William Shakespeare, ''Romeo agus Juliet'' (trans. of ''Romeo and Juliet''), Leabhar Breac, 2016. *Jonathan Swift, ''Camchuairt Ghuilivéir'' (trans. of ''Gulliver's Travels''), Leabhar Breac, 2016. *Hugo Pratt, ''Corto Maltese'' '''''Flag of Bones (Bratach na gCnámh) Series''''' Leabhar Breac published the Bratach na gCnámh series of books for young readers. Written in French by Alain Surget, illustrated by Annette Marnat and translated by Darach Ó Scolaí, this series uses the history of Caribbean Sea pirates<ref>{{Cite web|url=https://leabharbreac.com/en/product-category/alain-surget/|title=Alain Surget Archives|website=Leabhar Breac|language=en-US|access-date=2025-09-30}}</ref>: *Alain Surget, ''Éalú as Páras'' (''Escape from Paris''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''Oilean na Siorcanna'' (''Shark Island''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''Long na dTaibhsi'' (''Ship of the Ghosts''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''San Ochtapas Dubh'' (''In the Black Octopus''), Annette Marnat (Illustr.), Leabhar Breac, 2013. * Alain Surget, ''San Ionsai ar Veracruz'' (''The Attack on Veracruz''), Annette Marnat (Illustr.), Leabhar Breac, 2013. '''''For "First Readers" (children to 6 years old or so)''''' These books were written originally in Catalan by Spanish author Enric Lluch Girbés and translated into Irish by Ó ScolaÍ: *Enric Lluch, ''Ag Péinteáil an Tí'' (''Painting the House''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''An Colúr Bacach'' (''The Lazy Dove''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''An Phluais'' (''The Cave''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''Madra Dhaideo'' (''Grandpa's Dog''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''Fiacail Mháire'' (''Mary's Tooth''), Anna Clariana (Illustr.), Leabhar Breac, 2017. '''''Bruno Heitz''''' Leabhar Breac published a series of 3 Heitz books for small children. Published originally in French, this series of three comic books is about a blind mole named Cáitín Chaoch in Irish (and ''Louisette la taupe'' in French).<ref>{{Cite web|url=https://leabharbreac.com/en/product-category/bruno-heitz-en/|title=Bruno Heitz Archives|website=Leabhar Breac|language=en-US|access-date=2025-10-02}}</ref> Ó Scolaí translated these: *Bruno Heitz (author and illustr.), ''Práinneach'' (''Urgent''), Leabhar Breac, 2020 *Bruno Heitz (author and illustr.), ''Preab san Aer'' (''Bounce in the Air''), Leabhar Breac, 2020. '''''Books for Toddlers''''' Leabhar Breac has published 14 books written by French author Orianne Lallemand's and illustrated by Eleonore Thuillier, about Lallemmand's popular character Loup, Wolf. These are translated by Ó Scolaí: *Orianne Lallemand, ''An Mac Tire a Raibh Faitios an Domhain Air'' (trans. of ''The Son Who Saw the World in His Eyes''), Eleonore Thuillier  Illustr.), Leabhar Breac, 2018. *Orianne Lallemand, ''Macan agus an Goban'' (trans. of ''Macan and the Goblin''), Eleonore Thuillier (Illustr.), Leabhar Breac, 2018. * Orianne Lallemand, ''A Mac Tíre a Chuaigh go Tóin na Farraige'' (trans. of ''The Wolf Who Went to the Bottom of the Sea''), Éléanore Thuillier  (Illustr.), Leabhar Breac, 2019. '''''Board Books (for babies)''''' J. C. (Joan Carles) Girbés Aparisi is a Catalan author and editor. These books were written in Catalan and translated by Ó Scolai. *J. C. Girbés, ''An Phicnic'' (''The Picnic''), Silvia Ortega (Illustr.), Leabhar Breac, 2013. * J. C. Girbés, ''An Chóisir'' (''The Party''), Silvia Ortega (Illustr.), Leabhar Breac, 2013. *J. C. Girbés, ''Lá Mór Fada'' (''A Long Day''), Silvia Ortega (Illustr.), Leabhar Breac, 2014. *J. C. Girbés, ''Tabhair Leat do Leabhar'' (''Bring Your Book''), Silvia Ortega (Illustr.), Leabhar Breac, 2014. ==== Gradam Réics Carló ==== The Réics Carló prize is awarded for the best book in the Irish language for young readers. It is named for one of the characters of 20th-century writer [[wikipedia:Cathal_Ó_Sándair|Cathal Ó Sándair (Charles Saunders)]]. * ''An Bradán Feasa'' was “shortlisted for the Réics Carlo award 2010.”<ref name=":9" /> == References == {{reflist}} g0j3ldnb3qhh0jbuhe30wzwl29litx6 2844065 2844005 2026-11-02T23:18:36Z Scogdill 1331941 /* Overcoats */ 2844065 wikitext text/x-wiki {{Short description|Dress worn by Queen Victoria at her wedding to Prince Albert in 1840}} = Sandbox = Page to draft revisions for Wikipedia articles. For Gwladys Robinson, see Gwladys Lowther Robinson, [[Social Victorians/People/Ripon|Marchioness of Ripon]] and, earlier, [[Social Victorians/People/Lowther|Countess of Lonsdale]] ==References== {{reflist|2}} [[Category:1840 works]] [[Category:Royal wedding dresses|Victoria Queen]] [[Category:1840s fashion]] [[Category:British royal attire]] [[Category:Dresses in the Royal Collection of the United Kingdom|Victoria, Wedding]] [[Category:Diamond Jubilee of Queen Victoria]] = Victorian fashion = == Notes and Quotes == * Women followed Paris fashion in Europe and America, but London tailors led the way for men’s fashion. The Industrial Revolution actually impacted men’s fashion more than women’s. * Sewing machines in factories began mass producing cheap clothing for the largest group in the population, working class men. * Ready-made clothing affected the social structure, the economy, and gender differentiation. * Paper patterns, introduced in fashion magazines that targeted women, along with domestic sewing machines had more impact on women’s fashion than men’s. Haute couture was still bespoke (being individually constructed by tailors or dressmakers or fashion houses like Maison Worth). Sewing Machines and Clothes Before the industrial revolution wealthy people paid others to sew their clothes by hand and those who were of much lower economic status did all the manual labor necessary to feed and clothe the rich minority. The working class had to make all their own clothes. Sewing, however, came at the end of the process.  They began with raw wool cut from the backs of sheep or gathering flax. Only the wealthy could afford to buy fabric. They followed processes for cleaning and spinning the wool into thread and the processes of soaking and pulling out the fibrous strips to make into thread for linen fabric. After they had thread, it was tied to a loom so it could be woven into cloth. Then it had to be cut or torn into pieces. For many families scissors were a luxury. And the last step is sewing the pieces together to create a garment. Machines could not only sew, they could process the raw materials into thread and weave it into fabric much  quicker. Regarding clothing, working class men and their families benefitted greatly from ready-made clothes. Manufacturers marketed to them because they were the largest class. For example,<blockquote>–Men’s shirts, made by machine, take 3 hours and 10 minutes each to do; by hand, 15 hours and 20 minutes; a lady’s dress, by machine, takes 12 hours 4 minutes; by hand 20 hours 35 minutes; . . . a common frock-coat, by the machine occupies 17 hours 20 minutes; by the hand, 27 hours and 40 minutes: a linen vest, by machine, 3 hours 44 minutes; by hand 7 hours 18 minutes; a fine overcoat, by machine, 28 hours 13 minutes; by hand 31 hours 20 minutes; . . . <ref>Mitchell, Rebecca, ed. "22. 'Progress of the Sewing Machine,' ''Bow Bells'' (19 April 1865), p. 283." Section 6: Production and Industry. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.</ref></blockquote> === Industrial Revolution and Technological Advancements === * the railroad * mass production of fabrics for working-class men's clothing ** Jacquard looms * sewing machines * aniline dyes, including black === Expanding Middle Class === * gender roles getting more clearly defined and rigid (referring back to page overview) === French Revolution === Sarah Gharmallah Alzahrani and Safia Abdelaziz Saroukh (https://www.researchgate.net/profile/Safia-Saroukh/publication/385099194_The_Semiotic_Dimension_of_Men's_Fashion_in_Modern_Eras/links/671686fbd796f96b8ec4f90e/The-Semiotic-Dimension-of-Mens-Fashion-in-Modern-Eras.pdf):<blockquote>The 19th century started with a fashion landscape that was changing dramatically and rapidly from the styles of a generation earlier. The French Revolution brought fashions that had been emerging since the 1780s to the forefront. Neoclassicism now defined fashion as both men and women taking inspiration from classical antiquity. For women, the high-waisted silhouette in lightweight muslin was the dominant style, while fashionable men looked to the tailors of Britain for a new, refined look [17]. [17]  Franklin, H. (Aug 18, 2020). Published on Jun 25, 2020, Retrieved: <nowiki>https://fashionhistory.fitnyc.edu/1800-1809/</nowiki> 11/11/2023. Edited. ...</blockquote>Brent Shannon:<blockquote>"Costume," wrote Max Beerbohm in 1896, "enables us to classify any 'professional man' at a glance, be he lawyer, leech or who not" (24–25). A man's profession and class were read by his jacket, his hat, what he rode in, and how he carried himself. "Perhaps there is a tendency among Englishmen to judge a man too much by the shape of his hat or the kind of collar he wears," conduct author John Wanamaker confessed; "But one must remember that in England if you ''wear'' the wrong thing, you will probably ''do'' the wrong thing, and generally ''be'' the wrong thing" (1).¹¹ Such assertions were predicated on the powerful Victorian conviction that outward appearance reflected inner qualities.<ref>{{Cite book|title=The Cut of His Coat: Men, Dress, and Consumer Culture in Britain, 1860–1914|last=Shannon|first=Brent Alan|publisher=Ohio University Press|year=2006|location=Athens, Ohio}}</ref> (148)</blockquote> ==Women's fashion== == Men's fashion == (Two intros: one to Victorian Fashion, giving background for both women's and men's fashions; two to the decades for men's fashion) Among the influences on men's styles: * differentiation of genders (domestic ideology, ideology of the spheres) * the "Great Male Renunciation," the French Revolution and English tailoring * industrial revolution: mass production of fabrics and articles of clothing, especially for working-class men * sports and country clothing * Albert Edward, Prince of Wales, later King Edward VII, influence went beyond England; tailors led by him led fashion for the upper classes * then later, cult of youth * military uniforms * comfort and convenience * evolving definitions of gender, both femininity and masculinity evolved as concepts. The changing definitions of masculinity affected men's clothing. For the growing and rising middle classes, gender roles became more and more rigid around the concept of the separate spheres, the public for men and the private sphere for women. Cultural sources for men's clothing and men's fashion: industrial revolution; expanding middle class; technological advancements; an association in the culture between outward appearance and inward nature; French Revolution; Albert Edward, Prince of Wales Citation for Victorian Hell: <ref name=":29">{{Cite web|url=https://victorianhell.substack.com/p/victorian-mens-fashion-history-and|title=Victorian Men’s Fashion History and Clothing Guide|last="It's Monty, Actually"|date=31 October 2025|website=Victorian Hell|access-date=7 September 2026}}</ref> === Frock coats === * Apostol Mănescu, politician, wearing a frock coat, butterfly tie, black waistcoat with button chain and light coloured pants, drainpipe type. Circa 1866. Photo by Franz Duschek.jpg (1866)[[File:Apostol Mănescu, politician, wearing a frock coat, butterfly tie, black waistcoat with button chain and light coloured pants, drainpipe type. Circa 1866. Photo by Franz Duschek.jpg|thumb|Romanian (?) Politician, c. 1866, ill-fitting but probably handmade suit|center]] Full-length portrait of two young men wearing frock coats with sashes and pistols at their waists LCCN2006689564.tif (right)[[File:Full-length portrait of two young men wearing frock coats with sashes and pistols at their waists LCCN2006689564.jpg|thumb|1860, American, young men with frock coats, performers??|center]] * George Augustus Henry Sala.jpg (Mathew Brady, 1855–1865) * [[File:George Augustus Henry Sala.jpg|center|thumb|George Augustus Sala, 1855–1865, LoC]] ==== Dress and Conduct Manuals, Histories, Victorian Theory, Catalogs ==== # [Baker, William Henry.]  ''Clothes Dictionary''. Chicago, Ill.: Cahn, Wampold, 1901. https://commons.wikimedia.org/wiki/File:Clothes_dictionary;_(IA_clothesdictionar00bake).pdf. # [[c:File:Hint's_about_men's_dress,_right_principles_economically_applied_(IA_hintsaboutmensdr00barr).pdf|Hint's about men's dress, right principles economically applied (IA hintsaboutmensdr00barr).pdf]] (book, conduct manual, American, 1888) #[https://upload.wikimedia.org/wikipedia/commons/6/69/Fashion-_the_power_that_influences_the_world_%28IA_fashionpowerthat00foxg%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original Fashion- the power that influences the world (IA fashionpowerthat00foxg).pdf] (1871) (American, incl court dress, Victorian fashion theory, livery, "White, as it is the color of day, is expressive to us of the cheerfulness or gaiety which the return of day brings." [145], "Gentlemen's Furnishing and Goods," "Head and Foot Covering," Index) # [https://upload.wikimedia.org/wikipedia/commons/a/ae/How_to_behave_and_how_to_amuse._A_handy_manual_of_etiquette_and_parlor_games_%28IA_howtobehavehowto00sand%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original How to behave and how to amuse. A handy manual of etiquette and parlor games (IA howtobehavehowto00sand).pdf] (1895; American; Men's Dress, p. 23; besides conduct and dress, it has parlor games) # [https://upload.wikimedia.org/wikipedia/commons/8/8e/The_etiquette_of_men%27s_dress_%28IA_etiquetteofmensd00cluc%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original The etiquette of men's dress (IA etiquetteofmensd00cluc).pdf] (1888; American; some ads; anthology of individually written articles; lots of sports, outdoor events; search for "frock" — cutaway vs Prince Albert) #[https://upload.wikimedia.org/wikipedia/commons/e/e9/S.W._Silver_%26_Co.%27s_handbook_for_transatlantic_travellers_.._%28IA_swsilvercoshandb00silv%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original S.W. Silver & Co.'s handbook for transatlantic travellers .. (IA swsilvercoshandb00silv).pdf] (1883; American; lots of ads from SW Silver, with prices; has blank pages where ppl can write "Introductions" and "Engagements," but this is not a conduct book, it's travel guide / catalog to London) #[https://upload.wikimedia.org/wikipedia/commons/f/f6/How_we_go_to_those_who_cannot_come_to_us.._%28IA_howwegotothosewh00roge%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original How we go to those who cannot come to us.. (IA howwegotothosewh00roge).pdf] (Roger, Peet & Co.; American; catalog, spring 1891; pp. 11, clerical frock; 7 "Prince Alberts" double-breasted frocks; p. 23 shirts as underwear; has a thing on fitting and mail order; bathing suits, collars and cuffs; tennis shoes) == Victorian Men's Fashion (Intro to the Men's Fashion Section) == Museums and other historical sites have plenty of women’s clothes from the nineteenth century. There is great variety in both outer wear and underwear. “Men’s garments of the Victorian period have survived in far less quantity than women’s.” Buck p. 185 More waistcoats have survived than any other garment. The hardest garment to find are trousers. Very few survived the nineteenth century. Buck p. 185 By 1850 trousers dominated the fashion for men; the trouser straps under the shoe had disappeared; and a fly front had replaced the fall-front. Buck 185 It took most of the nineteenth century for men’s fashion to evolve into the coat, trousers and vest (waistcoat) that dominated the twentieth century and appears to be unchanging in the twenty-first century. “Men’s clothing, like women’s, was specific to each occasion and type of activity during the late nineteenth century.” Payne2, p. 518 For example, formal wear for social events was different from the casual, everyday wear for upper middle class and aristocratic men. Victorians also dressed in ensembles for hunting and riding (included knickerbockers–breeches that had a fuller cut). Casual and looser clothing was worn when participating in sports. Many different cuts of coats were available, like the Norfolk jacket, the morning coat, or the sack coat. “Tailored wool garments , originally inspired by the clothing of English country gentry, formed the backbone of men’s wardrobes in the early nineteenth century.” Men’s clothing began to change from styles that were colorful and flamboyant in the eighteenth century to styles that emphasized cut over color. By the end of the century men’s clothing had settled into a kind of uniform with coats and trousers usually made of the same fabric, often with a vest or waistcoat of the same fabric. The uniform included a shirt with a stiff collar, a necktie and very little jewelry. It took the entire nineteenth century to change to a rather static style that is still worn today. Ready-made clothing affected the social structure, the economy, and the gender differentiation. David Kutcha traces the origin of the 3-piece men's suit to King Charles II of England, who may be said to be wearing a coat, pants and a waistcoat, but otherwise is not wearing a "suit" we — or the Victorians — would recognize as such: <blockquote>... since 1666, male gentility has been associated with modesty and plainness in dress. Eschewing fashion as an increasingly feminized realm Charles II's vest inaugurated a new and essentially modern era of masculine aesthetics, one that reversed a long-held association between elaborate display and high social status. Manly thrift now displayed elite status.<ref>{{Cite book|title=The Three-Piece Suit and Modern Masculinity, England 1550–1850|last=Kutcha|first=David|publisher=University of California Press|year=2002|location=Berkeley and Los Angeles}}</ref> (2)</blockquote>So it's really Cromwell who should get credit?? The French Revolution caused both men and women to change stylistic direction radically. For one generation, stylish women wore simple white muslin dresses without foundation garments underneath. The changes in men's fashion was more lasting: men turned away from the ornamentation and ruffles of the ''ancien régime'', choosing instead understated and subtle tailoring to indicate good taste and wealth. Poole "comfort and convenience" — trousers, sack coats, collars and cuffs without a shirt, wool rather than silks and satins; knee breeches constantly riding up men as leaders of fashion, in spite of our sense that men didn't do fashion in the way that women did, were more serious than that === Dandy Stuff === Beau Brummell: "Men began to aspire to fine cutting, tailoring, and perfect fit in their clothes, flawless grooming and manners in themselves. The Englishman George Bryan (Beau) Brummell deserves much credit for the ideal of meticulous masculine appearance."<ref name=":25">{{Cite book|title=The History of Costume: From Ancient Mesopotamia Through the Twentieth Century|last=Payne|first=Blanche|last2=Winakor|first2=Geitel|last3=Farrell-Beck|first3=Jane|publisher=Addison-Wesley Longman|year=1992|isbn=0-06-047141-7|edition=2nd|location=New York, New York}}</ref> (458) "From 1796 to 1816, the Beau set the pattern for cleanliness and liberal use of starch." (459) === Gender Ideologies === '''Victorian gender ideologies''' relegated men to the "public sphere" and women to the "domestic sphere." (??) More than that “the cult of youth . . . the secularization of sport and the influence of modern warfare in generating men’s fashions, the tendency to prefer modern ideas of comfort and convenience, and the transformation of fashion knowledge and fashion urbanism from print to hyperreality”<ref name=":26">McNeil, Peter. "Men's Fashion: 1800–2022." Chapter 22. ''The Routledge History of Fashion and Dress, 1800 to the Present''. Routledge, 2024. https://opus.lib.uts.edu.au/bitstream/10453/182707/2/Men%27s%20Fashion%20200822_24_12_20_09_29_44.pdf DOI: 10.4324/9780429295607-27.</ref> (McNeil 1 or 411) changed the perceptions of masculinity and femininity. === Class === The growing middle class involved among other things more and more jobs and careers for young men as clerks, office workers, they were junior, subordinate, and they were commuting on the railroads from suburbs. Carolyn Kirby:<blockquote>In western Europe the fashion for plain dark suits coincided with the rise of the affluent middle-classes in a world where the pace of industrialisation and the globalisation of trade was accelerating as never before. The sharp, dark business suit became the last word in male power-dressing. And so it remains to this day.<ref name=":27">{{Cite web|url=https://historiamag.com/invent-masculine-fashion/|title=The invention of masculine fashion|last=Kirby|first=Carolyn|date=3 December 2025|website=Historia: Magazine of the Historical Writers' Association|access-date=25 August 2026}}</ref></blockquote> === Bertie === Albert Edward, Prince of Wales was very concerned with fashion and authoritative about it, with a very specific eye to small details. He was increasingly influential in men's fashion from the last third of the 19th and into the 20th century, and many noticed what he did and copied him.<blockquote>When that great lover of pleasure, Edward VII, visited Marienbad incognito as the Duke of Lancaster, he was followed by tailors from Paris, Budapest, Vienna, and Berlin who photographed him and took notes about his clothes. Edward VII introduced many [423–424] novelties into men’s fashion. ... He is also credited with making fashionable the creased trouser in 1909 (his groom dried them with a board weight after heavy rain, resulting in the line), turned-up cuff trouser (after hitching his trouser bottoms at a dirty racing track) and, as his girth grew, undoing the bottom button of his waistcoat."<ref name=":26" /> (424)</blockquote>Cowles<blockquote>The Prince was not just ‘a swell’. In the jargon of the day he was ‘a heavy swell’, and apparently there was a world of difference between the two terms. A swell was a rich young aristocrat who lived in extreme comfort; but a heavy swell added showmanship to the comfort and lived in a stylish luxury that even the French were obliged to envy. And of course the heavy swell was the acme of sartorial elegance. The Prince did not mind changing his dress half a dozen times a day. He loved clothes, and since whatever he chose to wear became the prevailing fashion overnight, he soon was regarded as an expert on the subject. His tailor-in-chief made a fortune. For many years he patronised a Mr. Poole. He discovered this gentleman by accident. He went to the theatre one night to see a well-known actor by the name of Fecher playing ‘Robert Macaire’. As an impecunious adventurer [129–130] Fechter was obliged to wear a coat that was torn and dirty, but Bertie’s expert eye noticed the elegant cut. At the end of the performance he asked Fechter for the name of his tailor, and Mr. Poole’s future was assured. The Prince had so many clothes he could never travel with less than two valets; and two more valets were left at home cleaning, brushing and pressing his vast wardrobe. There were suits and coats for every variation of every climate the world over. There were over a hundred pieces of headgear; and since Bertie was an honorary admiral and an honorary general of most of the countries of Europe, there was an entire room devoted to uniforms, sashes, epaulettes, belts, buckles, swords, feathers and other regalia. ... Men’s clothes became of such importance that new [130–131] shops sprang up like mushrooms in Savile Row, Clifford Street and Bond Street. Most of the Prince’s innovations were inspired by comfort and convenience. He altered the cut of the evening dress waistcoat, he shortened the tails on the tail coat, he left his frock coat open (due to an increasing girth), he introduced the black homburg, and he attended race meetings, not in the frock coat hitherto ''de regueur'' but in tweeds. He tried having his trousers creased down the sides rather than the front and back, in order to hide his bandy legs, but this idea did not catch on, and he soon discarded it himself. But the prince was not the only arbiter of men’s fashions. The band of  "heavy swells" who followed his lead gave him plenty of competition. Lord Raglan and Lord Petersham invented coats which are still named after them. Lord Dupplin the dinner jacket and Lord Cardigan the button-up sweater. But Lord Hardwicke made the most spectacular contribution. Men’s silk hats were made of beaver which was left in its original rough, shaggy state. Lord Hardwicke polished his hat until he could see his face in it, and consequently was known as "Glossy Top". He is responsible for the top hat as we know it today.<ref name=":28">{{Cite book|title=Gay monarch, the life and pleasures of Edward VII|last=Cowles|first=Virginia|publisher=Harper|year=1956|location=New York, New York|archive-url=https://archive.org/details/gaymonarchlifepl0000cowl/}}</ref> (129–131)</blockquote> === Men’s Clothing 1830–1900 (Intro to the Decades) === “Tailored wool garments , originally inspired by the clothing of English country gentry, formed the backbone of men’s wardrobes in the early nineteenth century.”<ref name=":25" /> (458) Men’s clothing began to change from styles that were colorful and flamboyant in the eighteenth century to styles that emphasized cut over color. By the end of the century men’s clothing had settled into a kind of uniform with coats and trousers usually made of the same fabric, often with a vest or waistcoat of the same fabric. The uniform included a shirt with a stiff collar, a necktie and very little jewelry. It took the entire nineteenth century to change to a rather static style that is still worn today. (this is middle-class men for day wear?) The silhouette for men sometimes followed the same line as women’s clothing on the top half of their bodies. For example, as women’s sleeves grew into the huge leg-of-mutton fullness, the shoulders of men’s coats also had added fabric to make a pouf at the shoulder, thus emphasizing a broad shouldered look across the front. Men’s silhouettes changed subtly without the exaggerated fullness of women’s skirts or the raising and lowering of the neckline. Once men began to wear trousers, they did not go back to formal knee breeches (except in European courts or for court wear for Victoria). The line of men’s garments changed in smaller increments than women’s. Coats and trousers might fit tighter in some decades and looser in others. They were slim, then boxy, flared, then straight and evolved into the contemporary line we see on men today. Men's fashions in the 19th century have been less researched than women's fashions, in part because of fewer artifacts of what men wore were saved in museums for study. Photography, the first century that we have actual images (rather than artists' interpretations) of what people wore. ==== Court and Formal Dress ==== Livery, working class Formal dress was separate attire, used for specified occasions, varies by decade ==== Shirts ==== Shirts and collars separated: "by 1827 detachable collars became available."<ref name=":25" /> (477) Shirts were considered more like underwear early in the century but by the end of the 1800s, they were becoming the outside garments popular in the twentieth and twenty-first centuries. Neck treatments (and shirt cuffs) Shirts were treated as underwear early in the century but by the end of the 1800s, they were becoming the outside garments popular in the twentieth and twenty-first centuries. ==== Coats and Jackets ==== The 1901 ''Clothes Dictionary'' says coats were "The principal garment of men's attire."<ref name=":30" />{{rp|10}} Several styles of coats were available to fashion-conscious aristocratic and upper-middle-class men. Men’s suits in this period were tailor made, fitted to one specific man. The factories, however, knew that the vast majority of men were working class and began to manufacture and sell ready-made shirts, coats and trousers. More clothes were available for working men and they cost less. The cheaper work clothes enabled upper-working-class men to own more than one set of clothes and to have more formal clothes for church or other events. Ready-made shirts were also available in generic sizes, small, medium and large. Generic sizing speeded up the mass production process and, since little of the shirts could be seen, they did not need custom fitting. Almost all of the coats Victorian men wore could be single or double breasted, with a couple of exceptions like morning coats and Norfolk jackets. On more formal occasions upper-class men wore a '''cutaway morning coat''' with light trousers during the daytime, and a dark '''tail coat''' and trousers in the evening. (define these terms) Morning coats was a general term for the jacket part of "full dress" or formal daytime wear, distinguished from formal evening wear. Most morning coats were cutaways, but the shape or cut is not the distinguishing feature: the bottom was curved or tailed, so the coat was short in front, it did not have a waist seam, single-breasted. Worn over a waistcoat and shirt. According to Payne, "During the early 1820s tailcoats were the most popular type of coat."<ref name=":25" />{{rp|474}} "Tailcoats gave way to skirted frock coats by the 1840s."<ref name=":25" /> (491) Tailcoats came down as far as the waist and then had tails (or "claw hammer" tails) attached at the waist in the back. Single-breasted, could be worn buttoned or unbuttoned in the front, over a waistcoat and shirt. Nowadays, the only tailcoats are tuxedos, a term not in use until the 1880s, and named for Tuxedo Park, a village in New York populated by the elite. Originally only the white coat, ''tuxedo'' gradually came to mean the entire suit. ==== Frock Coats ==== [[File:Frock coat MET CI38.23.39 F.jpeg|alt=Old-fashioned coat hanging on a mannequin|thumb|Frock coat, 1840, American]] Frock coats were introduced in the 1820s and worn throughout the nineteenth century with only minor changes to the line or fit. Sleeves were full at the top and waists were tight, creating an hourglass form. Initially frock coats were tightly fitted and closed with buttons. The more tailored frock coats had a waist seam and a flared skirt almost to the knees. The rate of fashion change notable in women's fashion was true for men's clothing as well. The frock coat persisted for a century, but it was in a state of constant (but subtle) evolution, with collars and lapels, coat length, the waist seam, the fullness of the coat itself as well as the sleeve, color, patterns like stripes or plaids. A Prince Albert is a double-breasted, very well-tailored, form-fitting frock coat.<ref name=":30">[Baker, William Henry.] ''Clothes Dictionary''. Chicago, Ill.: Cahn, Wampold, 1901. https://commons.wikimedia.org/wiki/File:Clothes_dictionary;_(IA_clothesdictionar00bake).pdf.</ref>{{rp|33}} Gentlemen, clergy, military Men around the world and especially in the British Empire wore frock coats: images and descriptions exist of men wearing frock coats in India<ref>Woodyatt, Nigel Gresley. ''Under Ten Viceroys; The Reminiscences of a Gurkha''. London: H. Jenkins Limited, 1922. Pp. 31, 107, 1774.</ref>, Iran (Persia at that time)<ref>Sparroy, Wilfrid. ''Persian children of the royal family; the narrative of an English tutor at the court of H. I. H. Zillus-Sultán, G.C.S.I''. London and New York, J. Lane, 1902. Pp. 11, 29, 30.[https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf].</ref>, Japan<ref>Watson, W. Petrie. ''Japan, Aspects and Destinies''. New York, Dutton: 1904. P. 7, 19, 31.</ref>, Egypt and Turkey<ref>Butler, Alfred Joshua. ''Court Life in Egypt''. London: Chapman and Hall, 1887. P. 7.</ref>, ... Payne: "In the 1720s, Englishmen adopted another coat [than the redingote]: a straighter, functional style that originated among the working classes. Made of plain woollen cloth in practical colors such as brown, it had no pleats, narrow sleeves, a small flat collar, and a minimum of decoration. This became known as the frock coat and was useful for riding, hunting, and overseeing one's lands." (402) ==== Trousers ==== Working-class men had been wearing trousers (or "trowsers") increasingly since the 16th century in Europe,('''citation''') but after the French revolution the middle and upper classes adopted them because "the safest dress was that of the working man, ... who did not wear knee breeches, but rather trousers."<ref name=":26" /> (416) Breeches: Men’s bifurcated garment that ended just below the knee Trousers (or long pants): men’s bifurcated garment that went down to the ankles or foot The most sweeping change in men’s fashion before 1820 was in the length of their pants. Trousers that went from waist to foot were so much more comfortable and popular than knee breeches that they replaced what had dominated men’s wear since the sixteenth century. And trousers have been worn for over two centuries now. They were originally tailored but began to be mass produced. Although the aristocracy were required to wear knee breeches at court, they wore trousers more than they wore breeches, and they could afford to have their suits and shirts hand sewn and individually. Working class men had to contend with small, medium, and large from mass production. This bifurcated garment was modified in specific places, like the length, the fullness over the hips or the width of the leg. In summer trousers were lighter in color — "grayish blue, aqua and pearl gray." (Payne 2, 505) In winter trousers mixed darker colors like "steel gray and brown, black and brown, and black and green." (Payne 505) More on summer colors, winter colors? [Find quote about QV's men's court dress and the men being so cold] Lewandowski: "First worn by the early horse-riding steppe people, introduced to Western world by Italians in 16th century. They were accepted as informal dress in the Directoire (1790–1815 C.E.). They became a part of semi-formal dress around 1815 and became worn commonly after 1830."<ref name=":31" /> (29 ==== Waistcoats ==== For most of the century, the waist-length sleeveless coat that men wore over their shirt and under their suit coat was called a waistcoat. By the 1880s, everyday waistcoats were called vests, and what was called a waistcoat was worn with formal, evening attire like tailcoats and tuxedos. Necklines rose and fell and collars and lapels came and went over the century, but by the end of the century, formal waistcoats had low necklines and lapels.<ref name=":25" /> (528) Many more waistcoats exist as artifacts in collections than vests because they were special, perhaps, and so they were saved.[[File:Waistcoat MET CI60.22.55 F.jpg|thumb|Men's waistcoat, 1850s]] Over the course of the century the waistcoat evolved from ornate and decorative to something simpler, whose fabric matched the suit coat and trousers — the vest, in fact. Victorian men put their creativity, color and specialty fabrics into their waistcoats. At the beginning of the century men's waistcoats were made from specialty fabrics, like velvet or brocade or a patterned silk. Early in the century, stylish men wore two vivid waistcoats at the same time.<ref name=":25" /> (474) An elaborate waistcoat might have a colorful, patterned base fabric as well as a floral ribbon embellishment (for example, see right). Long after coats and trousers were made from the same dark wool, waistcoats continued to be colorful, making them the focus of the ensemble. By the end of the 19th century, the plain style came to dominate men’s fashion (and continues to dominate it to this day). When waistcoats became vests, they became part of the plain style. * cc. 1830, dark brown or black w faces, gold buttons, standing collar: File:Waistcoat MET 38.68 F.jpg * c. 1830, beige and white woven fabric with rose buds stamped or embroidered??: File:Waistcoat MET 19.181.4 F.jpg * 1830–1835, dotted and floral stamped fabric, brown and tan: File:Waistcoat MET 1976.254.1.jpg * * c. 1840, flowered fabric: File:Waistcoat MET 1979.346.45 F.jpg * c. 1840, red, jacquard-like, double-breasted: File:Waistcoat MET 1979.346.47 F.jpg * 1840s, white brocade: File:Waistcoat MET 1980.132.8 F.jpg * 1840s, dark blue, red, green? stamped?: File:Waistcoat MET 1976.254.1.jpg * Mid 1840s, brown, embroidered: File:Waistcoat MET 24.160.3 F.jpg * late 1840s, embossed white with hash thingies: File:Waistcoat MET 1978.88.9 F.jpg * 1845 waistcoat made from blue Chinese silk damask: File:1845 waistcoat made from blue Chinese silk damask.jpg * 1845–1855, white brocade wedding waistcoat, piping on the edges: File:Waistcoat MET 1986.373.3 F.jpg * '''1849–1860''', black w gold lining, ribbon: File:Waistcoat MET 1978.477.25 F.jpg * * 1850, print of St. James's Park: File:Print (BM 1880,1113.2351).jpg * 1850s, white and blue, w blue floral design stamped on white-and-grey stamped on white fabric: File:Waistcoat MET 1978.477.28 F.jpg *c. 1853, brown and tan plaid, with red squares: File:Waistcoat MET 1984.90.2 F.jpg *1854: Fashion plates, groups of ppl: File:Paris, New York & Philadelphia fashions for spring and summer 1854, published & sold by F. Mahan, no. 211 Chestnut Street, Philadelphia. LCCN2003655770.jpg *c. 1855, French fashion plate, frock coats, waistcoats, etc.: File:Album z ilustracjami do czasopism z moda francuska, niemiecka i angielska. ca 1855 (9109644).jpg * 1857, cream, printed flowers on check, watch pocket: File:Waistcoat MET 1978.88.8 F.jpg * * c. 1860, dark floral? fabric with machine embroidery?: File:Waistcoat MET 1983.479.13 F.jpg * * late 1870s–early 1880s, gold, woven: File:Waistcoat MET 1979.34.1 F.jpg [[File:Homosexualitywilde.jpg|thumb|Oscar Wilde and Bosie Douglas, straw boater and derby, 1894]] ==== Overcoats ==== Overcoats or surcoats were, as their name suggests, coats worn for warmth and for protection from rain and snow on top of the other articles of clothng, including suit coats and frock coats. Overcoats became a significant fashion element and many styles were seen throughout the century. Victorians used the term overcoat or great coat, and they also specified what kind of overcoat: Chesterfield coat, Inverness coat, macintosh (or mackintosh), paletot, redingote, surtout, Ulster coat Surtout: "An overcoat cut in the style of a frock coat."<ref name=":30" /> (39) Lewandowski says something more generic, "term referring to any man's long cloak or coat."<ref name=":31">{{Cite book|title=The Complete Costume Dictionary|last=Lewandowski|first=Elizabeth J.|publisher=Scarecrow Press|year=2011}}</ref> (282) ===== Redingote ===== Redingotes had a rolled collar, sleeves with fullness at the shoulders and upper arm and narrow at the elbows to the wrist. It was fitted at the waist and reached mid-calf. existed various forms in the 18th century. Women's redingote 1865–1890, in France, to cover the bustle. (Lewandowski, 247–248) Payne: "The greatcoat or redingote evolved into an overcoat ..., but the overcoat lacked the pleats and large cuffs of the redingote." (402) high, double collar, standing collar, single- or double-breasted. "from riding coat" (Payne 402) Rebecca Mitchell: "full-length overcoat, usually open at the front and belted, popular in varying designs through the eighteenth and nineteenth centuries."<ref name=":32">Mitchell, Rebecca, ed. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.</ref> (527 or 565) [[File:George Stanhope, 7th Earl of Chesterfield.jpg|thumb|George Stanhope, 7th Earl of Chesterfield, wearing a Chesterfield coat, 1860]] ===== Chesterfield Coat ===== "Chesterfield coats were single-breasted, with fly fronts [regular buttons and button holes, as opposed to fall front, in which a panel of cloth covers the button holes and the placket covers the closure completely] and velvet collars."<ref name=":25" />  Payne2 p. 519 They were one kind of overcoat, named for George Stanhope, 6th Earl of Chesterfield. 1840s. Signature velvet collar. "Man's fitted overcoat with hidden buttons and velvet collar"<ref name=":31" /> (62) "fitted men’s knee-length overcoat, narrower than the frock coat and without a waist seam, popular from the end of the nineteenth century."<ref name=":32" /> (519 of 565) ===== Ulster Coat ===== ulster overcoat for travel. "Man's single- or double-breasted overcoat with waist belt. Initially had hood, but by 1870s was detachable. In 1875, the ticket pocket was added on the left sleeve above the cuff."<ref name=":31" /> (304) "Travel called for an ulster — a long, loose, heavy overccoat — or for an Inverness coat with attached cape."<ref name=":25" /> (527) "Heavy tweed ccoats for travel such as the ulster had half-belts and box pleating in back."<ref name=":25" /> (551) In 1872, George Darwin (Charles's son), in analyzing men's clothing as evolutionary behavior — like coloration for mating among birds — in the natural world, says, of trousers and Ulster coats:<blockquote>In both cases a form yields to a succeeding form, which is better adapted to the then surrounding conditions; thus, when it ceased to be requisite that men in active life should be ready to ride at any moment, and when riding had for some time ceased to be the ordinary method of travelling, knee breeches and boots yielded to trousers. The "Ulster Coat", now so much in vogue, is evidently largely fostered by railway travelling, and could hardly have flourished in the last century, when men either rode or travelled in coaches, where there was no spare room for any very bulky garment.<ref>Mitchell, Rebecca, ed. "3. 'George H. Darwin, ‘Development in Dress', ''Macmillan’s Magazine'' 26 (September 1872), pp. 410–16." Section 1: Fashion Theory in the Nineteenth Century. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.</ref> (125 of 565)</blockquote>"men’s overcoat with attached waist-length cape, popular in the nineteenth century."<ref name=":32" /> (530 of 565) ===== Inverness Coat ===== an Inverness coat with up to five cape collars and a high curved collar. Lewandowski: "(1840–1865 C.E.) to 20th century. Worn after 1858, man's large, loose, knee-length overcoat with fitted collar and long cape."<ref name=":31" /> (143) An Inverness coat had a high standing collar as well as up to 5 capes over a knee-length, loose overcoat.<ref name=":31" /> (143) In the 1880s, the Inverness was sometimes sleeveless, so the capes were not merely decorative. "men’s overcoat, with long, full, open sleeves, giving a cape-like appearance from the front; worn from the mid-nineteenth century to the early twentieth century as formal wear."<ref name=":32" /> (523 of 565) ===== Greatcoat ===== "large overcoat, commonly made of wool" (Lewandowski, 124) ===== Macintosh ===== "Introduced in 1836, short loose overcoat made of Mackintosh's patent India-rubber cloth. Came in drab or dark green."<ref name=":31" /> (181) "waterproofed overcoat, named after its inventor; now used generically to refer to raincoat."<ref name=":32" /> (524 of 565) ===== Paletot ===== "short, loose overcoat for men or women."<ref name=":32" /> (526 of 565) <ref name=":24" />Kay, Fiona and Neil R. Storey, ''Victorian Fashions for Women'', Pen & Sword History, Yorkshire - Philadelphia, 2022 ==== Hats ==== The upper-class continued to wear top hats, and [[bowler hats]] (known in America as a derby) were worn by the working class. 1830s This class of men also wore round cloth caps, often seen on male children. The straw boater was the most popular hat by the end of the century and was worn by all of the classes. Men wore [[top hats]], with wide brims in sunny weather. 1830s - 1900 Top hats briefly became the very tall "stovepipe" shape, but a variety of other hat shapes were popular. Homburgs, deerstalkers and “Trilbies” were also popular. Buck, p. 198-199 (George de Maurier novel ''Trilby'') Hats were ''de rigueur'' for all Victorians, all classes, all ages, all genders. ==== Shoes ==== [[File:Men's fashion silhouette of 1837.jpg|thumb|Men's coats, 1837]] [[File:Lev Wittgenstein by F. Kruger.JPG|thumb|Lev Wittgenstein by F. Kruger, 1836]] === 1830s and 1840s === For men's fashion, the 1830s and 1840s were dominated by a defined waist, lean limbs and tall hats. Working-class men had been wearing trousers for some time, but by the 1830s and '40s, middle- and upper-class men generally wore frock coats and trousers for casual or less formal attire in the city. In the country, what they wore depended on what they were doing — hunting, riding, shooting and so onen[[File:Man's Frock Coat LACMA M.2007.211.61 (1 of 4).jpg|alt=Male mannequin wearing a light-colored thigh-length coat similar to a suit jacket and a top hat|thumb|Men's frock coat, c. 1845]] [[File:Stockholms mode-journal- Tidskrift för den eleganta werlden 1847, illustration nr 2.jpg|thumb|Stockholm's fashion magazine, 1847]]Some men wore corsets during this time, so the waist was nipped in and the skirt of the frock coat flared out. The sleeves of their coats and jackets were slightly puffed at the shoulders. The trouser legs were straight with a strap under the instep. The figure on the left in the fashion plate from ''Modes de Paris Journal des Tailleurs'' (right) illustrates this silhouette very clearly as well as the frock coat, the strap under the instep, and the puffed shoulders. ==== Coats and Jackets ==== Frock coats had become fashionable for day wear in the 1820s<ref name=":25" />{{rp|474}} and were worn throughout the century. In the 1830s frock coats were tightly fitted and closed with buttons. The more tailored frock coats had a waist seam and a flared skirt almost to the knees. The main changes in frock coats in these two decades were in the colors and the neck treatments (collar and lapels). Initially frock coats were not necessarily black. They could be colorful — like "plum, mulberry, brown, bright blue, and bottle green" — with "contrasting velvet collars."<ref name=":25" />{{rp|474}} The truly bright, saturated colors made possible by synthetic dyes, however, were a couple of decades away still and were never used for frock coats. Middle- and upper-class men limited their use of tailcoats to evening wear and more formal occasions. (In the 1847 Stockholm fashion magazine, below right, the figure in the center is wearing a tailcoat.) A cutaway morning coat was worn with light trousers for formal events during the daytime, and a dark tail coat and trousers were worn in the evening. ==== Shirts ==== Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[cravats]] or neck ties. Before they were mass produced, shirts made of finer linen or cotton were tailored for upper and wealthy middle-class men. Depending on occupation, working-class men's shirts were made from a rougher, home-woven linen or cotton, often shaped loosely like a smock and sometimes without a front closure. Shirts were sometimes as long as knee-length. ==== Trousers ==== Men had adopted trousers over knee breeches for semi-formal dress "around 1815"<ref name=":31" /> (298) beginning after the French Revolution.<ref name=":26" /> (416) They were wearing trousers both for formal and semi-formal dress by the 1830s. Initially trousers "fit quite snugly ..., but in the 1840s they became looser."<ref name=":25" /> (477) Upper-class and aristocratic men still wore knee breeches at court and loose-fitting breeches (in the form of knickerbockers) for country wear and some sports, like riding and golf. Generally, however, they wore trousers more than they wore breeches. Men flocked to them once trousers became a stylish option because the looser fit was warmer in cold weather and more comfortable than the tight-fitting knee breeches from earlier in the century. Ready-made men's clothing was available in the US early, but by the 1840s ready-made suits were making an impact in Europe and the UK,<ref name=":26" /> (419–420) especially outside the upper classes. Aristocrats and upper middle class could afford the well-tailored cut and fit of individually hand-sewn trousers and coats. ==== Waistcoats ==== Waistcoats were both single- or double-breasted, with shawl or notched collars, and might be finished in double points at the lowered waist. Collars gradually lowered over the course of the two decades. Often the colorful vests were decorated with fur or velvet collars. “Waistcoats remained colorful, decorative items in men’s wardrobes after the rest of their costumes had become plain.” (Payne 2 460-462) <nowiki>:</nowiki> "During the 1820s and 1830s, men sometimes wore two waistcoats, in combinations like white velvet over rose-and-gold brocade."<ref name=":25" /> (474) ==== "Overcoats ==== Men wore a variety of topcoats or overcoats. Surtouts were overcoats built like long frock coats, like the one worn by Lev Wittgenstein in the portrait (above right). The redingote was most popular in the 1830s and 1840s. ==== Original Text ==== During the [[1840s in fashion|1840s]], men wore tight-fitting, calf length [[frock coat]]s and a [[waistcoat]] or vest. Sleeves were full at the top and waists were tight, creating an hourglass form. For more formal occasions, a cutaway morning coat was worn with light trousers during the daytime, and a dark tail coat and trousers was worn in the evening. Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[Cravat (early)|cravat]]s or neck ties. Men wore [[top hat]]s, with wide brims in sunny weather. === 1850s === The 1850s revealed several industrial inventions that significantly impacted clothing. First, the Industrial Revolution began (and still does today) with machines to weave fabric and machines to sew fabrics. Manufacturers quickly learned that the majority of the population was working class. Theirs were the simplest, least trimmed garments made in the nineteenth century. Second, mass production of shirts, trousers and coats changed how most people shopped for clothes. The aristocracy and upper middle class folks could still have their clothes made to specifically fit their bodies. Manufacturers knew that they could not mass produce clothing for each individual’s body. They invented generic sizing in small, medium, and large. They could, therefore, produce many more items of clothing with generic sizing. Third, Isaac Singer patented and began to manufacture home or domestic sewing machines (treadle machines) enabling more form-fitted '''dresses''' to be constructed for an individual’s specific shape and size. The fourth item to impact fashion and clothing was the invention of the paper pattern. These were introduced in fashion magazines targeting women. Almost all women were taught to sew by hand, whether they were aristocrats or working class. Sewing machines enormously speeded up the '''process of constructing from hours to minutes'''. Finally, aniline dyes made colors brighter and longer lasting. Prints and stripes, checks and plaids, fabrics had much more vibrant color with aniline dyes. The Industrial Revolution impacted the working class and changed their lives the most. children's clothes? Women made men's clothing too According to Judith Flanders, working-class men's clothes could be colorful. For example drivers of hansom cabs () dressed more up-scale, and drivers of hackneys () did not.<blockquote>While hackney drivers were also considered to be stereotypically shabby, hansom-cab drivers were generally represented as smartly dressed. A print in 1850 showed a driver in a snappy brown coat instead of the coachman’s heavy multiple-caped outfit, pale green striped trousers, short boots and top hat, the [167–168] reins held daintily in his gloved hands. Both cab and coach drivers wore top hats, but cabbies of a sporting bent later switched to bowlers, and in summer donned bright checked outfits.<ref name=":23">{{Cite book|title=The Victorian City: Everyday Life in Dickens' London|last=Flanders|first=Judith|publisher=Thomas Dunne Books|year=2012|location=New York, New York}}</ref> (167–168 [of 972])</blockquote> A variety of silhouettes was seen in the 1850s. Trousers were long and slim with a strap under the instep of the shoe. ==== Coats and Jackets ==== "Although the frock coat was worn throughout the century, it lost its dominant position in the 1850s, when the morning coat began to replace it."<ref name=":29" /> A jacket shorter than the frock coat or morning coat that had visible pockets began to be worn in about 1850. Cutaway tail coats were still reserved for formal occasions. All of these coats could be single or double breasted. They were called “ditto suits” by those who wore them because the coat, trousers, and waistcoat were all made from the same fabric. There were morning coats and cutaway morning coats; there were Norfolk jackets and ditto suits. The most formal were the cutaway tail coats for evening wear and balls; and the least formal were the boxy shaped sack coats. By the 1860s, it had become the most popular casual coat. ==== Shirts ==== During the [[1850s in fashion|1850s]], men started wearing shirts with high upstanding or turnover [[collar (clothing)|collars]] and [[necktie#Four-in-hand|four-in-hand necktie]]s tied in a bow, or tied in a knot with the pointed ends sticking out like "wings." ==== Trousers ==== Trousers had fly fronts and breeches were used for formal functions and knickers (short pants fuller cut than breeches) for horseback riding. Men's choice to wear trousers is an instance of the importance of comfort and convenience outweighing style and presentation, a value that increased in importance through the 20th century. ==== Waistcoats ==== Waistcoats remained the one garment that could be brightly colored and copiously trimmed on exotic fabrics. Waistcoats show that men were concerned with decorative elements in their ensemble. ==== Notes & Quotes ==== * transition from frock coats to ditto suits, 1850s (Payne, 463) ==== Hats ==== The upper-class continued to wear top hats, and [[bowler hat]]s were worn by the working class. Caps ==== Overcoats ==== Overcoats were cut fuller with wider sleeves and no extra fullness at the shoulder. Unlike the redingote, the overcoats had straight seams to the knee length. Many had wide lapels that reached below the waist. Working class men and their families benefitted more from the industrial revolution in regards to clothing.  Manufacturers marketed to them because because they were the largest class. [[File:Gazette of Fashion, February 1865, RP-P-2009-2675.jpg|thumb|1865, Frock coats, etc. (Paris)]] [[File:Gazette of Fashion, October 1864, RP-P-2009-2673.jpg|thumb|1864, Frock coats, etc., Paris]] [[File:Broadway fashions, New York LCCN2003677917.jpg|thumb|Butterick, Spring-Summer 1866, Broadway Fashions]] [[File:Four young men wearing frock coats - DPLA - 4706a21c9267c45034d51c8fc5309251 (page 1).jpg|thumb|1860s, working class?, ill fitting, frock coats, trousers, shirts, vest, derby hats]] === 1860s === [[File:Corporal frock coat 1862 med.jpg|thumb|Corporal frock coat 1862, 9-botton military]]The 1860s saw fewer inventions that influenced or changed the way people were acquiring clothes. But the changes begun by mass production continued with the working class. Little changed for the higher classes, except maybe paper patterns and sewing machines. The different shape and size in clothing continued to provide slightly different silhouettes for men in the upper and middle classes. As women's clothes grew more complex in cut and style, men's clothes became simpler in cut and style. ==== Coats and Jackets ==== Individualized fittings continued for upper class and wealthy middle class clients. (Working class as well as middle and upper class.) Working class was well over 60% of the population. For the largest class, ready-made garments were ideal for men and women who were working class. For the two upper classes morning coats still led the way but frock coats were still in demand and worn in business and more casual situations. Leisure wear like knickers and ditto suits became more popular in the 1860s. Haute couture wanted every occasion or event to have special clothing. In some ways men dressed more alike (coat, trousers, waistcoat) and in some ways there was more variety in the cut of their coats and in the manufactured clothing they could buy. Many upper class men still had their shirts custom made. Frock coats were shortened to knee-length and were worn for business, while the mid-thigh length [[sack coat]] slowly displaced the frock coat for less-formal occasions, with the overall effect of a looser silhouette. The Prince of Wales: "For the countryside such as at Sandringham, he permitted an informal dress code. The Henry Poole ledger marked as 'HRH 1865' is for an evening coat without tails, the first 'dinner jacket.'"<ref name=":26" /> (424) ==== Shirts ==== Ready-made shirts rarely were seen since shirts were more underwear than outerwear. Factories began to make and sell detachable collars and cuffs. It was more convenient to replace collars and cuffs each day than the time it took a servant (working class) to wash and starch enough to look like the celluloid many of those collars were made of. ==== Trousers ==== Trousers were cut full and boxy during most of the 1850s and 1860s. The strap under the shoe disappeared and trouser legs reached the back of the heel in length. Knickers were still worn for hunting or riding and breeches were still ''de rigueur'' at court. ==== Waistcoats ==== Waistcoats remained decorative and worn with most of the coats that were popular. The young “dandies” wore the most colorful and decorated waistcoats. These “dandies” gave their highest priority to their appearance. They were the avant garde of haute couture for men. ==== Necks Treatments ==== In the [[1860s in fashion|1860s]], men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin. ==== Hats ==== Top hats briefly became the very tall "stovepipe" shape, but a variety of other hat shapes were popular. ==== Overcoats ==== The single-breasted redingote continued in popularity among the aristocracy and upper middle-class.  These coats were often “in colors such as plum, mulberry, brown, bright blue, and bottle green . . . [with] contrasting velvet collars.” Payne2 p. 474 [[File:Mens Coats 1872 Fashion Plate.jpg|thumb|upright|Drawing of Victorian men 1870s]] [[File:Ashurst frock coat.jpg|thumb|Ashurst Frock coat]] [[File:Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat; Standing (fbc266c2feca43f6ad5b44e031635a8d).tif|alt=Old photograph of a man wearing a dark thigh-length coat similar to a suit coat with light-colored trousers|thumb|Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat, c. 1872]] [[File:Caillebotte Hugot.jpg|thumb|Portrait of Paul Hugot, 1878]] [[File:Oscar Wilde (1854-1900), by Hills & Saunders, Rugby & Oxford 3 april 1876.jpg|thumb|Oscar Wilde, 1876, ditto suit]] [[File:FrockCoat.jpg|thumb|English frock coat, 1870s]] === 1870s === To correct and prevent errors being made in court dress, in 1875 the Lord Chamberlain published ''Dress Worn by Gentleman at Her Majesty's Court'', "a summary of regulations for court uniform and dress."<ref>{{Cite book|url=https://www.google.com/books/edition/Dress_worn_by_Gentlemen_at_Her_Majesty_s/pvrbQCXq0MEC?hl=en|title=Dress worn by Gentlemen at Her Majesty's Court|last=Britain)|first=Victoria (Queen of Great|date=1875|language=en}}</ref> This is the kind of thing Bertie really cared about. Define court dress. A portly, boxy silhouette was sought for most of the 1870s. Older men whose stomachs had spread outward and thus looked prosperous to the Victorians were the ideal for men’s high fashion. Sack coats and trousers were cut a little wider in the legs. Corduroy for trousers and muslin for shirts was often used in working class clothing. ==== Coats and Jackets ==== There were minor changes to some of the coats Victorian men wore. Frock coats were shortened to the knee. Sack coats (or ditto coats — the forerunner of the suits worn by men in the western cultures of the twentieth and twenty-first centuries) were found to be so comfortable that most men of the upper and middle classes had a ditto suit. Men still wore morning coats and the aristocracy still wore cutaway coats with trousers to parties and balls but cutaway coats with knee breeches at court. ==== Shirts ==== Although ready-made shirts were available, the upper and middle classes still wore bespoke clothing hand-made, with better fabric and an excellent fit but stitched on machines. The suits and shirts made for the working class were of cheaper fabrics. ==== Trousers ==== Most men were wearing trousers  all of the time (except at court) and have worn them ever since. Trousers had button flies by the 1870s which most men found to be much more convenient than laces. Trousers did not “ride up” the leg in the way that knee breeches did when dancing or sitting. If a man did not have fashionable legs, they could be obscured behind trousers. ==== Waistcoats ==== White waistcoats began to be worn by men. Black were also worn when mourning. Changes in the waistcoat could be seen but were very subtle. For example, collars were lowered as the neckline grew deeper and deeper in front, revealing more of the fancy vest. ==== Overcoats ==== A longer version of the standard overcoat, “sometimes with fur collar, cuffs, and lining” was often worn for warmth. Payne2 p. 519 ==== Original Text ==== During the [[1870s in fashion|1870s]], three-piece suits grew in popularity along with patterned fabrics for shirts. Neckties were the four-in-hand and, later, the [[Ascot tie]]s. A narrow ribbon tie was an alternative for tropical climates, especially in the Americas. Both frock coats and sack coats became shorter and more form fitting. Flat straw boaters were worn when boating. [[File:Elliott & Fry10a.jpg|thumb|William Gladstone, 1887, morning coat]] [[File:British politicians gambling at Ascot Wellcome V0050378.jpg|thumb|British Politicians Gambling at Ascot, 1884]] [[File:Scribner's magazine (1887) (14595087238).jpg|thumb|1887, types of American servants]] === 1880s === The silhouette narrowed and straightened. That is, the line went straight down without the flare of skirts on the coats. This smooth line had few decorations. Men wore hardly any jewelry. They might have a boutonniere or a ring.  Suits were not  decorated. All coats were straighter and shorter. Depending on the time of day, the Prince of Wales might change his clothes (thus, his silhouette) up to six times per day,<ref name=":28" /> (129) and some aristocrats and upper-class men would follow his lead. ==== Coats and Jackets ==== Although the frock coat was worn throughout the nineteenth century, the single fabric sack suit was worn for casual wear and was beginning to be seen as a businessman’s suit.  Morning coats were still in use for some events as well as casual occasions. The modern tuxedo or Cowes jacket appeared and was worn for formal occasions. The tuxedo is still worn as formal dress today. ==== Shirts ==== Shirts were still custom made for the rich and the working class bought ready-made shirts. The shirt remained basically the same but it could handle the different neck treatments.  Even mass produced shirts could have buttons or button holes for detachable collars and cuffs. ==== Trousers ==== Trouser legs began to decrease in width as the coat became narrower and emphasized strong vertical lines. ==== Waistcoats ==== During the 1880s waistcoats were called vests [is it regional?] and much of their bold trim and colorful fabric disappeared and vests were constructed from the same fabric as the coat and trousers. The business suit for men developed in the 1880s and was firmly established in the 1890s. ==== Overcoats ==== Versions of topcoats increased, adding an ulster overcoat for travel and an Inverness coat with up to five cape collars and a high curved collar. Double breasted topcoats were popular and some were knee-length and some calf-length. ==== Original Text ==== During the [[1880s in fashion|1880s]], formal evening dress remained a dark tail coat and trousers with a dark waistcoat, a white bow tie, and a shirt with a winged collar. In mid-decade, the dinner jacket or [[tuxedo]], was used in more relaxed formal occasions. The [[Norfolk jacket]] and tweed or woolen breeches were used for rugged outdoor pursuits such as shooting. Knee-length topcoats, often with contrasting velvet or fur collars, and calf-length overcoats were worn in winter. Men's shoes had higher heels and a narrow toe. [[File:American fashions - The Jno. J. Mitchell Co. Litho., New York. LCCN2006676665.jpg|thumb|Fashion plate from the Jno. J. Mitchell Co., New York, 1899]] [[File:Alfred Milner Vanity Fair 1897-04-15.jpg|thumb|Alfred Milner, 1897, Leslie Ward ("Spy"), for Vanity Fair]] [[File:Nobl-08.jpg|thumb|Sherlock Holmes, 1892, Sidney Paget]] [[File:PUK1894page380rector poor.jpg|thumb|Rector and Drinker, 1894]] === 1890s === The dominant silhouette for men became the sack suit in the 1890s and while changes in the sack have been subtle, it remains the dominant silhouette today. ==== Coats and Jackets ==== “Sacks became ''the'' favored style of suit coat,”(Payne2, p. 532) especially among the younger men. Frock coats were associated with older men along with the cutaway coat. There were many more choices for sports wear and leisure wear than ever before. Norfolk jackets were often worn while hunting. Most informal styles could be purchased ready made. The rich still had their clothing custom made by tailors but even working class men could purchase most ready-made versions of the coats, trousers, and vests. ==== Shirts ==== Shirts could be made of patterned fabric as well as solid colors and white.  Shirts were beginning to be acceptable as outer wear especially when worn as sporting clothing. ==== Trousers ==== Early in the 1890s, “slim, straight trousers were news.” (Payne 2, p. 532) Trousers could be plaid or checked as well as the same fabric as the coat and vest. ==== Waistcoats ==== Fancy fabrics and bold trim were not seen much as the more subdued vest dominated the fashion scene. ==== Neck Treatments ==== In the 1860s, men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin. ==== Overcoats ==== “Topcoats included fingertip-length styles made of twill covert cloth. Single-breasted, fly-front overcoats were fashionable too. Roomy overcoats with fur linings and collars warmed the wealthy. Inverness overcoats remained in vogue.” Payne2, p.532 ==== Original Text ==== Starting from the [[1890s in fashion|1890s]], the [[blazer]] was introduced, and was worn for sports, sailing, and other casual activities.<ref>{{cite web|last=Landow|first=George|url=http://www.victorianweb.org/art/costume/90s/2.html|title=Men's informal sporting dress, late 1880s and '90s}}</ref> Throughout much of the Victorian era most men wore fairly short hair. This was often accompanied by various forms of facial hair including moustaches, side-burns, and full beards. A clean-shaven face did not come back into fashion until the end of the 1880s and early 1890s.<ref>{{cite web|url=http://www.victorianweb.org/art/costume/nunn21.html|title=Victorian Men's Fashions, 1850–1900: Hair}}</ref> Distinguishing what men really wore from what was marketed to them in periodicals and advertisements is difficult, as reliable records do not exist.<ref name="shannon597">{{cite journal|last=Shannon|first=Brent|title=Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914|journal=Victorian Studies|year=2004|volume=46|issue=4|pages=597–630|doi=10.1353/vic.2005.0022}}</ref> === Notes === *Men's suits buttoned higher up than today (Payne, 467) * Norfolk jackets and sack suits (Payne, 471) * formal attire, tuxedos with tails, cutaways (Payne, 469) * Keith Middlemas (https://archive.org/details/storyoffiesta00huxf/page/200/mode/2up?q=fashion) *Brent Shannon. "Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914." Victorian Studies 46, no. 4 (Summer 2004): 597–630. == Hats and headwear == [[File:Ford.madox.brown.last.emma.study.jpg|thumb|''Emma Hill'' by [[Ford Madox Brown]] (1853), a woman wearing a later version of the [[poke bonnet]]]] [[File:Hoed,_objectnr_KA_1237.tif|left|thumb|Perched bonnet style of the early 1870s.]] Hats were crucial to a respectable appearance for both men and women. === Men's Hats === The top hat, for example, was standard formal wear for upper- and middle-class men.[Payne] According to Blanche Payne, "The high top hat, usually black or dark gray, had reached its characteristic shape by 1798 and dominated the entire nineteenth century." (457–58) Although top hats were the dominant hat in the 19th century, other hats became popular for working classes and lower income middle class. “The style of an individual’s hat varied, depending on fashion and their social position, as well as their profession or chosen activity..” ''Goodman 53 of 460'' Other hats that became necessary and popular include the Derby, the straw Boater, and a flat cap with a short brim. In many cases the class, work activity and income could be determined by what kind of hat was on the head of the wearer. For some men, a hat supporting a particular sport or team was important. The Derby or Bowler hat was designed by William and Thomas Bowler, brother shopkeepers in 1849. ''(Goodman 55 of 460)'' It cost less than a top hat but  lasted longer and was soon worn by middle class bankers and clerks. Straw boaters were worn by the aristocracy for casual events and working class factory workers and agricultural laborers. By 1901 working class men had changed their preference to the flat caps which became the most popular hat for the workers. Headdress for men was an essential part of dress for the entire period of the Victorian age, from the 1830s through the end of the century. Judith Flanders describes the hats worn by men in London, <blockquote>It is difficult to bear in mind the importance of hats as not only markers of class and income, but also as indicators of respectability. [509–519] [George Augustus] Sala commented that "every" man throughout the history of the world "must, necessarily and habitually, wear some kind of covering to his head". Postmen wore hats, small children wore hats, field labourers and market gardeners wore hats, cricketers, skaters — all sportsmen — wore hats. It was, self-evidently, impossible to go outdoors without one. ... Those in professional occupations wore pot hats, as did clerks and all those with pretensions to middle-class status. Even doctors' delivery boys wore battered hand-me-down pot hats: "the nap rusty, the band a mournful strip of tarnished lace; but still a Hat", which "stamps him as being associated, in however slender a manner, with a learned profession". Cloth caps were for labourers, for costers and for boys. ... Artisans wore caps made out of paper, which they folded [510–511] themselves and so could easily replace as they became dirty.<ref name=":23" /> (509–511 [of 972]) </blockquote>A pot-hat is a kind of derby or bowler in men's hats. (Lewandowski, 237) ==== Original Text ==== Hats were crucial to a respectable appearance for both men and women. The top hat, for example, was standard formal wear for upper- and middle-class men.<ref name=":4">{{Cite book |last=Steele |first=Valerie |url=https://archive.org/details/fashioneroticism0000stee |title=Victorian Fashion. Fashion and Eroticism: Ideals of Feminine Beauty from the Victorian Era to the Jazz Age |publisher=Oxford University Press |year=1985 |isbn=978-0-19-503530-8 |pages=[https://archive.org/details/fashioneroticism0000stee/page/51 51]–84 |url-access=registration}}</ref> For women, the styles of hats changed over time and were designed to match their outfits. === Women's Hats === For a discussion of the history of plumes and feathers, see [[Social Victorians/Victorian Things#Ostrich Plumes and Prince of Wales's Feathers|Ostrich Plumes and Prince of Wales's Feathers in ''Victorian Things'']]. ==== Original Wikipedia Text ==== During the early Victorian decades, hats were modest in size and design, straw and fabric bonnets being the popular choice. [[Poke bonnet]]s, which had been worn during the late [[Regency period]], had high, small crowns and brims that grew larger until the 1830s, when the face of a woman wearing a poke bonnet could only be seen directly from the front. They had rounded brims, echoing the rounded form of the bell-shaped hoop skirts. Bonnets shrunk at the end of the 1860s and moved to a perched position in the early 1870s as hairstyles grew in scale and intricacy. This led to the popularization of hats, which became the headwear of choice for the remainder of the Victorian era.<ref name="g4223">{{cite book |last=Cunnington |first=Cecil Willett |title=English Women's Clothing in the Nineteenth Century |date=1990-05-01 |publisher=Courier Corporation |isbn=0-486-26323-1 |publication-place=New York |page=}}</ref> [[File:The_London_and_Paris_ladies'_magazine_(Apr_1885)_03.png|thumb|Flower pot style hat of 1885.]] The 1880s saw a hat inspired by the top hat for women known as the flowerpot hat, and the 1890s saw the popularity of the boater. The hats of the late Victorian era were covered with elaborate creations of silk flowers, ribbons, and above all, exotic plumes; hats sometimes included entire exotic birds that had been stuffed. Many of these plumes came from birds in the Florida everglades, which were nearly made entirely extinct by overhunting. By 1899, early environmentalists like [[Adeline Knapp]] were engaged in efforts to curtail the hunting for plumes. By 1900, more than five million birds a year were being slaughtered, and nearly 95 per cent of Florida's shore birds had been killed by [[Plume hunting|plume hunter]]s.<ref>{{cite web|title=Everglades National Park|url=https://www.pbs.org/nationalparks/parks/everglades/|archive-url=https://web.archive.org/web/20090927085907/http://www.pbs.org/nationalparks/parks/everglades/|url-status=dead|archive-date=27 September 2009|publisher=PBS|access-date=7 November 2011}}</ref> == Shoes == The women's shoes of the early Victorian period were narrow and heelless, in black or white satin. By 1850s and 1860s, they were slightly broader with a low heel and made of leather or cloth. Ankle-length laced or buttoned boots were also popular. From the 1870s to the twentieth century, heels grew higher and toes more pointed. Low-cut pumps were worn for the evening.<ref name=":4" /> == Cosmetics == [[Victorian-era cosmetics]] were typically less obvious than ours. However, small amounts of pale face powder or powdered blush were widely used.<ref>{{Cite book |last=Goodman |first=Ruth |title=How to be a Victorian |date=2014 |publisher=Penguin Books |isbn=978-0-670-92136-2 |location=London}}</ref> Tints were sometimes added to face creams. Some cosmetics contained toxic or caustic ingredients like lead, mercury, ammonia, and arsenic {{Citation needed|date=October 2025}}. Hair color ==Mourning black== {{See also |Mourning stationery}} [[File:The royal children in mourning Mar 1862.jpg|thumb|Victoria's five daughters (Alice, Helena, Beatrice, Victoria and Louise), photographed wearing mourning black beneath a bust of their late father, Prince Albert (1862)]] [[File:Mourning dress MET 50.40.3a-b front CP4.jpg|alt=Black Victorian mourning dress|thumb|Mourning Dress, 1894–95]] In Britain, black is the colour traditionally associated with mourning for the dead. The customs and etiquette expected of men, and especially women, were rigid but evolving during much of the Victorian era. The expectations depended on a complex hierarchy of close or distant relationship with the deceased. (Davidoff) The closer the relationship, the longer the mourning period and the wearing of black. The wearing of full black was known as First Mourning, which had its own expected attire, including fabrics, and an expected duration of 4 to 18 months. Following the initial period of First Mourning, the mourner would progress to Second Mourning, a transition period of wearing less black, which was followed by Ordinary Mourning, and then Half-mourning. Some of these stages of mourning were shortened or skipped completely if the mourner's relationship to the deceased was more distant. Half-mourning was a transition period when black was replaced by acceptable colours such as lavender and mauve, possibly considered acceptable transition colours because of the tradition of [[Church of England]] (and [[Catholic Church|Catholic]]) clergy wearing lavender or mauve [[Stole (vestment)|stoles]] for funeral services, to represent the [[Passion (Christianity)|Passion of Christ]].<ref>{{cite web|title=The Colors of the Church Year|url=http://fullhomelydivinity.org/articles/colors.htm|publisher=Consortium of Country Churches|access-date=6 November 2011|archive-date=13 November 2011|archive-url=https://web.archive.org/web/20111113075214/http://fullhomelydivinity.org/articles/colors.htm|url-status=dead}}</ref> The mourning dress worn by Queen Victoria (below, right) "shows the traditional touches of mourning attire, which she wore from the death of her husband, Prince Albert (1819–1861), until her own death."<ref>{{Cite web|url=https://www.metmuseum.org/art/collection/search/155839?&searchField=All&sortBy=Relevance&deptids=8&ft=queen+victoria&offset=0&rpp=20&amp;pos=2|title=Mourning Dress, 1894–95|last=The Metropolitan Museum of Art|date=7 September 2019|website=The Metropolitan Museum of Art|access-date=7 September 2019}}</ref> Dating from 1894–95, Queen Victoria wore this dress as a result of the death of the eldest son of the Prince and Princess of Wales, Eddy, in line to the throne. === Norms for mourning=== ''Manners and Rules of Good Society, or, Solecisms to be Avoided'' (London, Frederick Warne & Co., 1887) gives clear instructions, such as the following:<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–83}}</ref> {| class="wikitable" |- ! Relationship to deceased !! First mourning !! Second mourning !! Ordinary mourning !! Half-mourning |- | Wife for husband || 1-year, 1-month; [[bombazine]] fabric covered with [[Crape|crepe]]; [[widow's cap]], [[lawn cuff]]s, collars || 6 months: less crepe || 6 months: no crepe, silk or wool replaces bombazine; in last 3 months jet jewellery and ribbons can be added || 6 months: colours permitted are grey, lavender, mauve, and black-and-grey |- | Daughter for parent || 6 months: black with black or white crepe (for young girls); no linen cuffs and collars; no jewellery for first 2 months || 4 months: less crepe || – || 2 months as above |- | Wife for husband's parents || 18 months in black bombazine with crepe || – || 3 months in black || 3 months as above |- | Parent for son- or daughter-in-law's parent || – Black armband in representation of someone lost || – || 1-month black || – |- | Second wife for parent of a first wife || – || – || 3 months black || – |} The complexity of these etiquette rules extends to specific mourning periods and attire for siblings, step-parents, aunts and uncles distinguished by blood and by marriage, nieces, nephews, first and second cousins, children, infants, and "connections" (who were entitled to ordinary mourning for a period of "1–3 weeks, depending on level of intimacy"). Men were expected to wear mourning black to a lesser extent than women, and for a shorter mourning period. After the mid-19th century, men would wear a black hatband and black suit, but for only half the prescribed period of mourning expected of women. Widowers were expected to mourn for a mere three months, whereas the proper mourning period expected for widows was up to four years.<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–9}}</ref> Women who mourned in black for longer periods were accorded great respect in public for their devotion to the departed, the most prominent example being Queen Victoria herself. It was not uncommon for a widow who did not remarry to wear half-mourning for the rest of her life, except when another death necessitated full mourning. For example, Alexandra, Princess of Wales wore half-mourning for the rest of her life after her eldest son Eddy died in 1894. Empress Elisabeth of Austria did the same, as did Empress Eugénie of France. They reverted to full mourning when appropriate, but they never wore less than half-mourning after their sons' deaths. Women with lesser financial means tried to keep up with the example being set by the middle and upper classes by dyeing their daily dress. Dyers made most of their income during the Victorian period by dyeing clothes black for mourning.<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|page=341}}</ref> == Technological advancement == The technological changes that affected the manufacture and consumption of clothing in the Victorian age included the following: * the mass production of fabrics — for example, "by the early 1850s there were thousands of steam-powered looms churning out millions of miles of fabric every year"  [62] * the invention of aniline dyes, which were much more vibrantly colored and resistant to fading than the natural dyes that had been used. — . Invented by chemist [[William Henry Perkin]] in 1856, the first aniline dye mauveine (or mauve) "wash[ed] the fashionable landscape in a haze of purple."<ref name=":22">{{Cite book|title=The Dress Diary: Secrets from a Victorian Woman's Wardrobe|last=Strasdin|first=Kate|publisher=Pegasus Books|year=2023|location=New York, New York}}</ref> (247) Other intense and, to the Victorians, intensely exciting colors followed, but the new synthetic additions to fabric sometimes included chemicals harmful to their wearers. For example, a "bright-magenta hue was achieved by adding arsenical-based chemicals to existing aniline dyes, brightening the already luminous shades – but these left residues themselves, along with a toxic labour trail in their wake."<ref name=":22" /> (255) Perhaps the most famous of these is arsenic green, used on fabrics, wallpapers, and trim: "The craze for artificial foliage to adorn the heads and dresses of women of fashion in the mid-nineteenth century had seen the proliferation of flower workshops, where young women in their hundreds laboured to produce the lifelike green leaves and blooms that would make a fetching headdress or would trail becomingly across the bodice of a gown. The lushness of the green was achieved by the application of a powder, a pigment that was created by mixing copper and the highly toxic chemical, arsenic trioxide. The physical effects of working with this poisonous compound were horrific. Contemporary medical drawings depict the green hue of the skin and dreadful open lesions on the hands of the maker, whilst the daily gradual ingestion of the powder by the flower girls was eventually fatal."<ref name=":22" /> (255) * the invention of a sewing machine that could be used in the home. Although sewing machines were already in use in the clothing industry, in 1858 Isaac Merritt Singer began to sell "lightweight domestic machines" for home sewing, radically increasing women's control over their own dress.<ref name=":24">{{Cite book|title=Victorian Fashions for Women|last=Kay|first=Fiona|last2=Storey|first2=Neil R.|publisher=Pen & Sword History|year=2022|isbn=978 1 39900 416 9|location=Yorkshire and Philadelphia|pages=}}</ref> (91 [of 298]) * the spread of journalism for women and fashion journalism * the Jacquard loom: "French inventor Joseph-Marie Jacquard’s loom attachment mechanized the weaving of figured fabrics by 1804."<ref name=":25" /> (454) Perhaps not at the same scale as these but as important in 1850s designs was a technology that turned iron into steel, which could then be drawn into fine wires.<ref name=":3">{{Cite book|title=The Culture of Fashion|last=Breward|first=Christopher|publisher=Manchester University Press|year=1995|pages=145–180}}</ref> Steel was refined to a malleable state so that thin blades could be curved into concentric circles (called hoops) and connected with wires to form the cage. Technological advancements not only influenced the economy but brought a major change in the fashion styles worn by men and women. As the Victorian era was based on the principles of gender, race and class.<ref>{{cite journal|last1=Graham|first1=P|title=The Victorian Era|url=https://archive.org/details/in.ernet.dli.2015.261548|journal=Digital Library of India}}</ref> Much advancement was in favor of the upper class as they were the ones who could afford the latest technology and change their fashion styles accordingly. In 1830s there was introduction of horse hair crinoline that became a symbol of status and wealth as only the upper-class women could wear it. In 1850s there were more fashion technological advancements hence 1850s could rightly be called a revolution in the Victorian fashion industry such as the innovation of artificial cage crinoline that gave women an artificial hourglass silhouette without layers of petticoats, which was lighter and more hygienic.<ref>{{cite book|last1=Shrimpton|first1=J|title=Victorian Fashion|publisher=Bloomsbury Shire Publications}}</ref> Synthetic dyes, such as [[mauveine]] (aniline purple), were introduced in 1856, adding bright colours to garments. In 1855's ''[[Haute couture]]'' was introduced as tailoring became more mainstream in years to follow.<ref>{{cite book|last1=Aspelund|first1=Karl|title=Fashioning Society|publisher=Fairchild Books}}</ref> Charles Frederick Worth, a prominent English designer, became popular amongst the upper class though its city of destiny always is Paris. Haute couture became popular at the same time that sewing machines were invented.<ref name="Haute Couture">{{cite book|last1=Martin|first1=Richard|last2=Koda|first2=Harold|title=Haute Couture|publisher=The Metropolitan Museum of Art}}</ref> Princess [[Eugénie de Montijo|Eugenie]] of France wore the Englishman dressmaker, Charles Frederick Worth's couture and he instantly became famous in France though he had just arrived in Paris a few years ago. In 1855, Queen Victoria and Prince Albert of Britain welcomed [[Napoleon III]] and Eugenie of France to a full state visit to England. Eugenie was considered a fashion icon in France. Queen Victoria, who had been the fashion icon for European high fashion, was inspired by Eugenie's style and the fashions she wore.{{Citation needed|date=October 2025}} Later, Queen Victoria also appointed Charles Frederick Worth as her dress maker and he became a prominent designer amongst the European upper class. Charles Frederick Worth is known as the father of the haute couture as later the concept of labels were also invented in the late 19th century as custom, made to fit tailoring became mainstream.<ref>{{cite book|last1=Saillard|first1=Olivier|last2=Zazzo|first2=Anne|title=Paris Haute Couture|publisher=Skira Flammarion}}</ref> By the 1860s, when made-to-fit tailoring was popular in Europe, crinolines were considered impractical. In the 1870s, women preferred more slimmer silhouettes, hence bodices grew longer and the polonaise, a skirt and bodice made together, was introduced. In 1870s the Cuirass Bodice, a piece of armour that covers the torso and functions like a corset, was invented. Towards the end of Victoria's reign, dresses were flared naturally as crinolines were rejected by middle-class women. Designers such as Charles Frederick Worth were also against them. All these inventions and changes in fashion led to women's liberation as tailored looks improved posture and were more practical.<ref name="Haute Couture"/> dressmakers, couturiers, modistes == Home decor == {{main|Victorian decorative arts}} Home decor started spare, veered into the elaborately draped and decorated style we today regard as Victorian, then embraced the retro-chic of [[William Morris]] as well as pseudo-[[Japonaiserie]]. == Myths and Oversimplifications == === Modesty === {{main|Victorian morality}} {{Original research|section|date=May 2008}} [[File:1868-skirt-lengths-girl-ages-Harpers-Bazar.gif|thumb|upright|"The proper length for little girls' skirts at various ages", from ''[[Harper's Bazaar]]'', showing a 1900 idea of how the hemline should descend towards the ankle as a girl got older]]Many myths and exaggerations about the period persist to the modern day. Examples include the idea of men's clothing is seen as formal and stiff, women's as elaborate and over-done; clothing covered the entire body, and even the glimpse of an ankle was scandalous. Critics contend that [[corset]]s constricted women's bodies and lives. Homes are described as gloomy, dark, cluttered with massive and over-ornate furniture and proliferating [[bric-a-brac]]. Myth has it that even piano legs were scandalous, and covered with tiny [[pantalette]]s. === Tight Lacing === Tight-lacing, which was not possible until the development of the grommet in 1828, was famously controversial in the Victorian age, generating many column inches of profitable newspaper copy, in part because it was (and still is) fetishistic and subversive in that adolescent girls used it as a means of rebellion and upper-working- or lower-middle-class shop girls saw it as a means of upward mobility.<ref name=":21">{{Cite book|title=Fashion and Fetishism: Corsets, Tight-Lacing and Other Forms of Body-sculpture|last=Kunzle|first=David|publisher=History Press|year=2013|isbn=978 0 7524 9545 3|location=Stroud, Gloucestershire|pages=}}</ref> (71 [of 1182]) No evidence exists that tight lacing was widespread or particularly dangerous.<ref name=":21" /> () In truth, men's formal clothing may have been less colourful than it was in the previous century, but brilliant [[waistcoat]]s and [[cummerbund]]s provided a touch of colour, and [[smoking jacket]]s and [[robe|dressing gown]]s were often of rich Oriental [[brocade]]s. This phenomenon was the result of the growing textile manufacturing sector, developing mass production processes, and increasing attempts to market fashion to men.<ref name="shannon597"/> Corsets stressed a woman's sexuality, exaggerating hips and bust by contrast with a tiny waist. Women's [[evening gown]]s bared the shoulders and the tops of the breasts. The [[jersey dress]]es of the 1880s may have covered the body, but the stretchy novel fabric fit the body like a glove.<ref>{{cite book |last=Gernsheim |first=Alison |title=Victorian & Edwardian Fashion: A Photographic Survey |year=1981 |publisher=Dover Publications |location=New York |page=65|edition=New |isbn=0-486-24205-6}}</ref> Home furnishing was not necessarily ornate or overstuffed. However, those who could afford lavish draperies and expensive ornaments, and wanted to display their wealth, would often do so. Since the Victorian era was one of increased social mobility, there were ever more ''[[nouveaux riches]]'' making a rich show. The items used in decoration may also have been darker and heavier than those used today, simply as a matter of practicality. London was noisy and its air was full of [[soot]] from countless coal fires. Hence those who could afford it draped their windows in heavy, sound-muffling curtains, and chose colours that didn't show soot quickly. When all washing was done by hand, curtains were not washed as frequently as they might be today. There is no actual evidence that piano legs were considered scandalous. Pianos and tables were often draped with [[shawl]]s or cloths—but if the shawls hid anything, it was the cheapness of the furniture. There are references to lower-middle-class families covering up their [[pine]] tables rather than show that they couldn't afford [[mahogany]]. The piano leg story seems to have originated in the 1839 book, ''A Diary in America'' written by Captain [[Frederick Marryat]], as a satirical comment on American prissiness.<ref>{{cite book |last1=Marryat |first1=C.B. |title=A Diary in America: With Remarks on Its Institutions |date=1839 |publisher=Longman, Orme, Brown, Green, and Longmans |location=London, England |volume=2 |pages=246–247 |url=https://books.google.com/books?id=2-VEAAAAIAAJ&pg=PA246}} From pp. 246-247: "I was requested by a lady to escort her to a seminary for young ladies, and on being ushered into the reception-room, conceive my astonishment at beholding a square piano-forte with four ''limbs''. However, that the ladies who visited their daughters, might feel in its full force the extreme delicacy of the mistress of the establishment, and her care to preserve in their utmost purity the ideas of the young ladies under her charge, she had dressed all these four limbs in modest little trousers, with frills at the bottom of them!"</ref> Victorian manners may have been as strict as imagined—on the surface. One simply did not speak publicly about sex, childbirth, and such matters, at least in the respectable middle and upper classes. However, as is well known, discretion covered a multitude of sins. Prostitution flourished. Upper-class men and women indulged in [[adultery|adulterous]] liaisons. == Gallery == {{gallery |2=A mid-Victorian interior: ''Hide and Seek'' by [[James Tissot]], c. 1877 Image:Winterhalter Elisabeth.jpg|3=Dress designed by [[Charles Frederick Worth]] for [[Elisabeth of Bavaria|Elisabeth of Austria]] painted by [[Franz Xaver Winterhalter]].|4=File:Frith A Private View detail.jpg|5=[[William Powell Frith]]'s painting of 1883 contrasts women's [[Aesthetic dress]] (left and right) with fashionable attire (center).|6=File:Tissot lilacs 1875.jpg|7=Day dress, c. 1875 [[James Tissot]] painting.|8=File:James Abbot McNeill Whistler 011.jpg|9=[[James McNeill Whistler|Whistler]]'s [[Portrait of Lady Meux]], 1882 Image:Jeanna_Samary-Renoir.png|10=[[Pierre-Auguste Renoir|Renoir]]'s portrait of [[Jeanne Samary]] in an [[evening gown]], 1878|11=File:Melville_-_Queen_Victoria.jpg|12=Portrait by [[Alexander Melville (artist)|Alexander Melville]] of [[Victoria of the United Kingdom|Queen Victoria]], 1845|13=File:Henry Treffry Dunn Rossetti and Dunton at 16 Cheyne Walk.jpg|14=An artistic interior: [[Dante Gabriel Rossetti]] reading to [[Theodore Watts-Dunton]] in the drawing room at No. 16 [[Cheyne Walk]], 1882|15=File:Punch - Masculine beauty retouched1.png|16=Men's swimwear: Cartoon from ''[[Punch (magazine)|Punch]]'' by [[George du Maurier]]}} == See also == * [[Emily Clapham]] * [[Victorian decorative arts]] * [[Victorian dress reform]] * [[Victorian morality]] * [[Victoriana]] * [[Women in the Victorian Era]] * [[Charles Frederick Worth]] === Time periods === * [[1830s in fashion]] * [[1840s in fashion]] * [[1850s in fashion]] * [[1860s in fashion]] * [[1870s in fashion]] * [[1880s in fashion]] * [[1890s in fashion]] === Women's clothing === * [[Corset]] * [[Corset controversy]] * [[Tightlacing]] * [[Bloomers (clothing)|Bloomers]] * [[Bodice]] === Contemporary interpretations === * [[Steampunk]] * [[Neo-Victorian]] * [[Lolita Fashion|Lolita]] == References == {{Reflist}} == Further reading == *{{cite book |author=Phipps, Elena| title= ''From Queen to Empress: Victorian dress 1837-1877'' | location=New York | publisher=The Metropolitan Museum of Art | year=1988 | isbn=0870995340| url= http://libmma.contentdm.oclc.org/cdm/compoundobject/collection/p15324coll10/id/69547/rec/235 | display-authors=etal}} * Sweet, Matthew – ''Inventing the Victorians'', St. Martin's Press, 2001 {{ISBN|0-312-28326-1}} == External links == * [http://www.victorians.co.uk/victorian-fashion Victorian Fashion] {{Webarchive|url=https://web.archive.org/web/20180407223711/http://www.victorians.co.uk/victorian-fashion |date=7 April 2018 }} * [https://www.victorianvoices.net/topics/fashion/index.shtml VictorianVoices.net] – Fashion articles and illustrations from Victorian periodicals; extensive fashion image gallery * [http://www.cracked.com/article_19575_5-ridiculous-sex-myths-from-history-you-probably-believe.html Victorian myths] * [http://www.victorianstation.com/lifestylemenu.htm Victorian fashion, etiquette, and sports] {{Webarchive|url=https://web.archive.org/web/20180103162620/http://www.victorianstation.com/lifestylemenu.htm |date=3 January 2018 }} * [http://www.thesmartset.com/article/article12180701.aspx Background on "A Diary in America"] * [http://www.mccord-museum.qc.ca/en/keys/webtours/VQ_P2_17_EN.html Form and Fashion] — the evolution of women's dress during the 19th century (many photographs) * [http://www.mccord-museum.qc.ca/en/keys/games/jeu2/ Educational Game: Mix and Match] — build a 19th-century dress using a virtual mannequin * {{cite web |publisher= [[Victoria and Albert Museum]] |url= http://www.vam.ac.uk/content/articles/v/victorian-dress-at-v-and-a/ |title= Victorian Dress |work= Fashion, Jewellery & Accessories |date= 14 January 2011 |access-date= 2011-04-03}} *[http://cv.vic.gov.au/stories/creative-life/fashion-detective-fashion-fiction-and-forensics/ Fashion detective: Fashion, Fiction and Forensics in nineteenth century Australian fashion] on Culture Victoria {{Timeline of clothing and fashion|state=collapsed}}{{Victorian era|state=collapsed}} [[Category:Victorian fashion| ]] [[Category:19th-century fashion|*]] [[Category:1900s fashion]] [[Category:History of Western fashion]] [[Category:19th century in the arts]] =From ''Women in the Victorian era''= ===Victorian women's fashion=== {{Multiple issues|{{tone|date=March 2023}} {{more footnotes needed|date=March 2023}}|section=y}}{{Further|Victorian fashion}} The ideal Victorian woman was pure, chaste, refined, and modest. This ideal was supported by etiquette and manners. The etiquette extended to the pretension of never acknowledging the use of undergarments (sometimes generically referred to as "unmentionables"). The discussion of such a topic, it was feared, would gravitate towards unhealthy attention on anatomical details. As one Victorian lady expressed it: "[those] are not things, my dear, that we speak of; indeed, we try not even to think of them", in contrast to current norms.<ref>{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=20}}</ref> The pretence of avoiding acknowledgement of anatomical realities met with embarrassing failure on occasion. In 1859, the Hon. Eleanor Stanley wrote about an incident where the [[Louisa Cavendish, Duchess of Devonshire|Duchess of Manchester]] moved too quickly while manoeuvring over a [[stile]], tripping over her large [[hoop skirt]]: {{blockquote|[the Duchess] caught a hoop of her cage in it and went regularly head over heels lighting on her feet with her cage and whole petticoats above, above her head. They say there was never such a thing seen – and the other ladies hardly knew whether to be thankful or not that a part of her undergarments consisted in a pair of scarlet tartan [[knickerbockers (clothing)|knickerbockers]] (the things Charlie shoots in) which were revealed to the view of all the world in general and the [[Aimable Pélissier|Duc de Malakoff]] in particular".<ref>{{cite book|last=Cunnington|first=C. Willett|title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations|year=1990|publisher=Dover Publications|isbn=978-0-486-26323-6|pages=20–1}}</ref>}} However, despite the fact that Victorians considered the mention of women's undergarments in mixed company unacceptable, men's entertainment made great comedic material out of the topic of ladies' [[bloomers (clothing)|bloomers]], including men's magazines and music hall skits.<ref>{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=22}}</ref> Victorian women's clothing followed trends that emphasised elaborate dresses, skirts with wide volume created by the use of layered material such as [[crinoline]]s, hoop skirt frames, and heavy fabrics. Because of the impracticality and health impact of the era's fashions, a [[Victorian dress reform|dress reform movement]] began among women. The ideal silhouette of the time demanded a narrow waist, which was accomplished by constricting the abdomen with a laced [[corset]]. While the silhouette was striking, and the dresses themselves were often exquisitely detailed creations, the fashions were cumbersome. At best, they restricted women's movements and at worst, they had a harmful effect on women's health. Physicians turned their attention to the use of corsets and determined that they caused several medical problems: compression of the thorax, restricted breathing, organ displacement, poor circulation, and prolapsed uterus.<ref name="O'Connor"/> Articles advocating the reform of women's clothing by the British National Health Society, the Ladies' Dress Association, and the [[Rational Dress Society]] were reprinted in ''The Canada Lancet'', Canada's medical journal. In 1884, Dr J. Algernon Temple of Toronto even voiced concern that the fashions were having a negative impact on the health of young women from the working classes. He pointed out that a young working-class woman was likely to spend a large part of her earnings on fine hats and shawls, while "her feet are improperly protected, and she wears no flannel petticoat or woollen stockings".<ref name="O'Connor"/> [[File:Bloomers.jpg|thumb|1850s illustration of a woman wearing [[bloomers]]]] [[Florence Pomeroy]], Lady Haberton, was president of the Rational Dress movement in Britain. At a National Health Society exhibition held in 1882, Viscountess Haliburton presented her invention of a "[[divided skirt]]", which was a long skirt that cleared the ground, with separate halves at the bottom made with material attached to the bottom of the skirt. She hoped that her invention would become popular by supporting women's freedom of physical movement, but the British public was not impressed by the invention, perhaps because of the negative "unwomanly" association of the style with the American [[Bloomers]] movement.<ref>{{cite book|last=Murray|first=Janet Horowitz|title=Strong-Minded Women and Other Lost Voices from 19th Century England|year=1982|publisher=Pantheon Books|location=New York|isbn=0-394-71044-4|pages=[https://archive.org/details/strongmindedwome00jane/page/68 68–70]|url=https://archive.org/details/strongmindedwome00jane/page/68}}</ref> [[Amelia Jenks Bloomer]] had encouraged the wearing of visible bloomers by feminists to assert their right to wear comfortable and practical clothing, but it was no more than a passing fashion itself among radical feminists. The movement to reform women's dress would persist and have long-term success, however; by the 1920s, [[Coco Chanel]] was successful at selling a progressive, far less restrictive silhouette that abandoned the corset and raised hemlines. The new silhouette symbolised modernism for trendy young women and became the 20th century standard. Other Paris designers continued reintroducing pants for women and the trend was gradually adopted over the next century. Fashion trends, in one sense, travelled "full circle" over the course of the Victorian era. The popular women's styles during the [[Georgian era]], and at the very beginning of Victoria's reign, emphasized a simple style influenced by flowing gowns worn by women in [[Ancient Greek clothing|Ancient Greece]] and [[Clothing in ancient Rome|Rome]]. The [[Empire waist]] silhouette was replaced by a trend towards ornate styles and an artificial silhouette, with the restrictiveness of women's clothing reaching its low point during the mid-century passion for narrow corseted waists and hoop skirts. The iconic wide-brimmed women's hats of the later Victorian era also followed the trend towards ostentatious display. Hats began the Victorian era as simple [[Bonnet (headgear)|bonnets]]. By the 1880s, milliners were tested by the competition among women to top their outfits with the most creative (and extravagant) hats, designed with expensive materials such as silk flowers and exotic plumes such as ostrich and peacock. As the Victorian era drew to a close, however, fashions were showing indications of a popular backlash against excessive styles. Model, actress and socialite [[Lillie Langtry]] took London by storm in the 1870s, attracting notice for wearing simple black dresses to social events. Combined with her natural beauty, the style appeared dramatic. Fashions followed her example (as well as Queen Victoria's wearing of mourning black later in her reign). According to [[Harold Koda]], the former Curator-in-chief of the [[Costume Institute at The Met|Metropolitan Museum of Art's Costume Institute]],<ref>{{cite web|url=http://www.metmuseum.org/about-the-museum/press-room/exhibitions/2014/death-becomes-her|title=Death Becomes Her: A Century of Mourning Attire : October 21, 2014-February 1, 2015|website=Metmuseuim.org|access-date=7 November 2021}}</ref> "The predominantly black palette of [[mourning]] dramatizes the evolution of period silhouettes and the increasing absorption of fashion ideals into this most codified of etiquettes," said Koda, "The veiled widow could elicit sympathy as well as predatory male advances. As a woman of sexual experience without marital constraints, she was often imagined as a potential threat to the social order." ====Evolution of Victorian women's fashion==== <gallery> File:Fashion plate December 1844.jpg|Ladies' December Fashions (1844). Hand-coloured steel engraving from a women's magazine. File:Thegalleryofhmscalcutta james tissot 1876.jpg|''[[The Gallery of HMS Calcutta]]'' by [[James Tissot]] (1876). [[Bustle]]s were fashionable in the 1870s and 1880s. File:Mrs lillie langtry george frederic watts 1880.jpg|''Mrs. Lillie Langtry'' by [[George Frederic Watts]] (1880). File:Five-women-on-queenslander-steps-r.jpg|Fashionable women in [[Queensland]], Australia around 1900. </gallery> {{Short description|Irish writer (born 1963)}} {{Use Irish English|date=August 2025}} {{Use dmy dates|date=August 2025}} {{Infobox writer | name = Darach Ó Scolaí | image = Darach Ó Scolaí.JPG | alt = Man holding prize-winning book | caption = Ó Scolaí in 2019 | birth_name = Darach Ó Scolaí | birth_date = {{Birth date and age|1963|df=y}} | birth_place = [[County Galway]], The Republic of Ireland | death_date = | death_place = | occupation = Writer, artist, publisher | alma_mater = [[University of Galway]] | years_active = 1998–present | genre = Novel, retelling, translation, play, screenplay, illustrated book for children and adults | other_names = | spouse = | children = 3 | awards = [[Awards and Honors received by Darach Ó Scolaí|Awards and Honors]] | signature = | website = }}[[File:Darach Ó Scolaí.JPG|thumb|Darach Ó Scolaí, holding ''Oileán an Órchiste'' (his translation of Robert Louis Stevenson's ''Treasure Island'')]] == Darach Ó Scolaí == Darach Ó Scolaí (<small>Irish:</small> [/ˈda.rax/ /oː/ /sˠkˠoː/l̪ˠəi/]; born 1963<ref>{{Cite web|url=https://portraidi.ie/en/darach-o-scolai/|title=Darach Ó Scolaí|date=20 October 2017|website=Portráidí (Portraits of Irish-Language Writers)|access-date=1 August 2025}}</ref>) is an Irish author who works in a number of genres, from novels, plays and screenplays to illustrated books for children and adults. He began his literary career in 1998 writing screenplays, stage plays, retellings and translations; he began to publish novels in 2008. Ó Scolaí is widely recognized as a leading figure in contemporary Irish literature, known as “one of the most important Irish language writers of his generation”<ref>{{Cite journal|last=Poirtéir|first=Cathal|date=2022|title=? Suil an Daill: Constant Tensions and Shifting Allegiances|url=https://booksirelandmagazine.com/suil-an-daill-constant-tensions-and-shifting-allegiances/|journal=Books Ireland}}</ref> and "one of the great Irish language novelists [duine d’úrscéalaithe móra na Gaeilge]."<ref name=":17" /> His writing has been called “the high literature of the Irish language.”<ref>Ó Coimín, Maitiú. ''Nós'' 2 February 2018). Qtd. in "Táin Bó Cuailnge." ''Leabhar Breac''. Retrieved 25 August 2025.</ref> Much of his fiction is based on a knowledge of traditional Irish tales and narrative practices as well as Irish history. He specializes in literary and [[wikipedia:Historical_fiction|historical fiction]], or as novelist Alan Titley says, Ó Scolaí’s “peak (for now), or at least his greatest imaginative interest, is the historical novel [tá an chuma air gurb é a bhuaic (go fóill), nó ar a laghad, a mhórspéis samhlaíochta, an t-úrscéal staire].”<ref name=":11">{{Cite journal|last=Titley|first=Alan|date=Fall 2020|title=An Stíl Go Deo!: Soather Dharach Uí Scolaí (The style would be forever!: Worker Darach Ó Scolaí)|url=https://www.jstor.org/stable/27046090|journal=Comhar|volume=80, No. 10|pages=27|via=JSTOR}}</ref> His retellings of old stories and tales from their original Middle and Early-Modern Irish into Modern Irish ([[wikipedia:Irish_language|Gaeilge]]) are respected for their accessibility to students and language learners as well as for their artistry. Ó Scolaí also regularly reviews books and lectures and writes on literature and culture. Beyond his writing, Ó Scolaí is a publisher and has co-produced a number of film, television shows and stage plays. == Life == Ó Scolaí was born in Dublin and raised in the Galway [[wikipedia:Gaeltacht#Galway Gaeltacht|Gaeltacht]] (Irish-speaking) regions of Cois Fharraige on the north shore of Galway Bay, in the Republic of Ireland, where he lives now with his wife and children in Lochán Beag (Indreabhán).<ref name=":7">{{Cite journal|date=30 October 2024|title=Duais don úrscéal liteartha is fearr buaite ag Darach Ó Scolaí ag Oireachtas na Samhna|url=https://tuairisc.ie/duais-don-ursceal-liteartha-is-fearr-buaite-ag-darach-o-scolai-ag-oireachtas-na-samhna/|journal=Tuairisc}}</ref><ref>{{Cite journal|last=Ní Scolaí|first=Aifric|date=2024|title=Darach Ó Scolaí|url=https://www.taiscecf.ie/ealaiontoiri?category=Scr%C3%ADbhneoir|journal=Taisce Chois Fharraige}}</ref> He graduated the [[wikipedia:University_of_Galway|University of Galway]] (then University College Galway) with a B.A. in 1983.<ref>{{Cite web|url=https://www.linkedin.com/in/darach-ó-scolaí-20026920/|title=Darach Ó Scolaí|last=Ó Scolaí|first=Darach|date=August 2025|website=LinkedIn}}</ref> === Writing and Publishing === Ó Scolaí writes in Irish ([[wikipedia:Irish_language|Gaeilge]]), his native language, and lives in an area defined for the predominant presence of Irish as the vernacular language, the language spoken at home. Irish was the language of his parents' home and is the language of children as well. He is fluent in Irish and English and conversant in French. None of his works has been translated into English. ==== Leabhar Breac ==== In 1995 Darach Ó Scolaí and his brother Caomhán Ó Scolaí — a [[wikipedia:Typography|typographer]] and designer — founded the publishing house Leabhar Breac at Indreabhán (Inverin), County Galway. Their father “Séamas Ó Scolaí was an editor at An Gúm and worked on the Irish-English dictionary team [bhí a n-athair Séamas Ó Scolaí ina eagarthóir sa Ghúm agus d’oibrigh sé ar fhoireann an fhoclóra Gaeilge-Béarla].”<ref name=":0">{{Cite web|url=https://leabharbreac.com/en/about-us/|title=About Us|date=2024|website=Leabhar Breac|access-date=1 July 2025}}</ref> Darach Ó Scolaí has been publisher and literary editor at Leabhar Breac since its founding. Named for [[wikipedia:An_Leabhar_Breac|An Leabhar Breac (The Speckled Book)]], Leabhar Breac publishing house has more than 140 books in print.<ref name=":0" /> Leabhar Breac aims to publish Irish-language books that meet “a high literary and artistic standard.”<ref name=":0" /> Besides the content, Leabhar Breac is known for the typically "superb [thar cionn]" quality of the design and production of the "physical book [leabhar fisiciúil]."<ref name=":8">{{Cite journal|last=Ní Mhuilneoir|first=Gráinne|date=30 July 2024|title=‘Bláthnaid’ – leabhar álainn i sraithín álainn faoi mhná|url=https://tuairisc.ie/blathnaid-leabhar-alainn-i-sraithin-alainn-faoi-mhna/|journal=Tuairisc}}</ref> Its books regularly win awards for literary and artistic quality. Leabhar Breac also publishes translations for children and adults from various early versions of Irish as well as from French and English (and has published translations of books for young readers from Spanish, Catalan, and Italian as well). Leabhar Breac prints its books in Ireland. === Stage and Screen === ==== Rosg ==== In 1998 along with Ciarán Ó Cofaigh,<ref name=":1">{{Cite web|url=http://www.rosg.ie/en/about/History_6/|title=About Us: History|date=July 2025|website=Rosg|access-date=1 August 2025}}</ref> Ó Scolaí co-founded the film and television production company [http://www.rosg.ie/en/ Rosg] and was co-director until 2006. Rosg produced Ó ScolaÍ’s films ''Cosa Nite'' (1999), ''An Leabhar'' (2001) and ''Na Cloigne'' (2010). He left Rosg in 2006 to devote his time to other artistic activities. ==== Ealaín ar Oileán ==== In 2004, along with Val Balance, Ó Scolaí co-founded the annual artists' symposium Ealaín ar Oileán (trans., Art on an Island). The Irish-language symposium was held annually in the Áras Éanna arts and cultural center on Inis Oírr ([[wikipedia:Inisheer|Inisheer]], the smallest of the [[wikipedia:Aran_Islands|Aran Islands]]) from 2004 to 2013. Ó Scolaí was its co-director from its founding<ref>{{Cite web|url=https://ga.wikipedia.org/wiki/Darach_Ó_Scolaí.|title=Darach Ó Scolaí|date=3 February 2024|website=Vicipéid|access-date=1 July 2025}}</ref> until 2013. Besides being its co-director, Ó Scolaí has taken part in this conference as an artist<ref>{{Cite journal|date=16 January 2005|title=Darach Ó Scolaí|url=https://web.archive.org/web/20050116163252/http://bliainiris.com/authors/darach_oscolai.html|journal=Bliainiris}}</ref> and writer<ref name=":2">{{Cite web|url=http://ealainaroilean.ie/ealainaroilean.html|title=The Conference|date=7 September 2013|website=Ealaín ar Oileán|archive-url=https://web.archive.org/web/20130907083744/http://ealainaroilean.ie/ealainaroilean.html|archive-date=7 September 2013|access-date=1 August 2025}}</ref>. ==== Salamandar ==== In 2006 Ó Scolaí founded the stage production company Salamandar and directed his own play ''An Braon Aníos''. His plays ''An tSeanbhróg'' (2009) and ''Craos'' (2008) were also produced by Salamandar.<ref name=":19">{{Cite web|url=https://leabharbreac.com/en/product-category/darach-o-scolai/|title=Darach Ó Scolaí|date=2024|website=Leabhar Breac|access-date=1 July 2025}}</ref> == Works == === Novels === * [[wikipedia:An_Cléireach|''An Cléireach'' (trans., ''The Clerk'')]], Leabhar Breac, 2007. The Oireachtas Prize for Literary Fiction, 2007; The Ó Súilleabháin Award (Book of the Year) in 2008, and "named as ‘the best novel since the turn of the Century’ by Comhar."<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-cleireach/|title=An Cléireach - Leabhar Breac - Irish language novel|website=Leabhar Breac|language=en-US|access-date=2025-10-24}}</ref> * ''Na Comharthaí'' (trans., ''The Signs''), Leabhar Breac, 2014. * ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. The Oireachtas Prize for Literary Fiction, 2019.<ref name=":4">{{Cite web|url=https://leabharbreac.com/en/shop/fiction/suil-an-daill/|title=Súil an Daill|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Bódléar'', Leabhar Breac, 2024. The Oireachtas Prize for Literary Fiction, 2024<ref name=":7" />; The Ó Súilleabháin Award (Book of the Year) in 2025; featured in the 2025 Listen-Up Irish Summer Challenge for students of the Irish language.<ref>{{Cite news|url=https://connachttribune.ie/novel-approach-helps-people-learn-irish-in-a-creative-way/|title=Novel approach helps people learn Irish in a creative way|last=Murphy|first=Judy|date=3 October 2025|work=Connaught Tribune|access-date=24 October 2025}}</ref> === Retellings, Translations and Editions === The retellings and translations are into modern Irish. * ''Feis Tigh Chonáin'' (trans., ''The Feast of Conán's House''), Leabhar Breac, 2000; a retelling of a 15^th-century tale from the [[wikipedia:Fenian_Cycle|Fenian Cycle]].<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/feis-tigh-chonain/|title=Feis Tigh Chonáin|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''An Ceithearnach Caolriabhach'' (trans., ''The Narrow-Striped Kern''), Leabhar Breac, 2002; a retelling from c. 1500, also illustrated by Darach Ó ScolaÍ.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-ceithearnach-caolriabhach/|title=An Ceithearnach Caolriabhach|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), Leabhar Breac, 2017, both a modern edition of an 11th-century epic and an annotated edition.<ref name=":3">{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/tain-bo-cuailnge-2-2/|title=Táin Bó Cuailnge|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> "''Táin Bó Cuailnge'' won the Aodán Mac Poilín Memorial Prize 2017."<ref name=":19" /> * ''Deirdre'', Leabhar Breac, 2023, a “picture book for adults” with artist Anastasia Melnykova.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/deirdre/|title=Deirdre|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Part of the [[wikipedia:Ulster_Cycle|Ulster Cycle]], ''Deirdre'' is a retelling of the story of possibly the most widely known Irish figure from the early tales and sagas.<ref>{{Cite book|title=A Dictionary of Celtic Mythology|last=MacKillop|first=James|publisher=Oxford University Press|year=2004|isbn=9780198609674|pages=181}}</ref> * ''Bláthnaid'', Leabhar Breac, 2024, a “picture book for adults” with artist Anastasia Melnykova; “one of the great stories of the [[wikipedia:Ulster_Cycle|Ulster Cycle]].”<ref name=":4" /> * ''Sadhbh,'' Leabhar Breac, 2025, a picture book for adult readers, illustrated by Alé Mercado; a retelling of the medieval tale ''Ceasacht Inghine Ghuile (''trans., ''The Complaint of Guile's Daughter'').<ref name=":5">{{Cite web|url=https://leabharbreac.com/en/tales-of-wonder/|title=Tales of Wonder|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Eoghan Béal'', Leabhar Breac, 2025, a picture book for adult readers illustrated by Alé Mercado<ref name=":5" />; a retelling of the medieval tale ''[https://ga.wikipedia.org/wiki/Caithr%C3%A9im_Cellaig Cathréim Ceallaigh]'' from ''The Yellow Book of Leacan.''<ref name=":5" /> === For Young Readers === Ó Scolaí has written illustrated books for young readers (8–10 years old) in two series, the Fionn Series and the Scéalta Staire series, and translated a large number of classics and popular books for children of all ages. The number of these written and translated works suggests a commitment to children and their literacy in Irish. The Fionn Series “is a retelling ... of the great legends of the Fianna for the young Irish readers of today.”<ref name=":6">{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/doiteoir-na-samhna/|title=Dóiteoir na Samhna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> [[wikipedia:The_Boyhood_Deeds_of_Fionn|Macgnímartha Finn (The Boyhood Deeds of Fionn)]] is a medieval story in the [[wikipedia:Fenian_Cycle|Fenian Cycle]]. * ''An Bradán Feasa'' (trans., ''The Salmon of Knowledge''), Leabhar Breac, 2010, “shortlisted for the Réics Carlo award 2010.”<ref name=":9">{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/an-bradan-feasa/|title=An Bradán Feasa|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Dóiteoir na Samhna'' (trans., ''The Halloween Burner''), 2010.<ref name=":6" /> * ''Bodach an Chóta Lachna'' (trans., ''The Churl in the Dun Coat''), 2011.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/|title=Bodach an Chóta Lachna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> The Scéalta Staire (Historical Stories) series<ref name=":9" /> * ''Mánas Ó Dónaill'', 2000. * ''Seán Ó Néill'', Leabhar Breac, 2000. * ''Gráinne Mhaol Ní Mháille'', Leabhar Breac, 2003. * ''Tadhg Dall Ó hUiginn'', Leabhar Breac, 2003. ==== Translations ==== * Robert Louis Stevenson, ''Oileán an Órchiste'' (trans. of ''Treasure Island''), Leabhar Breac, 2014.<ref>{{Cite journal|date=2025-06-19|title=Oireachtas na Gaeilge|url=https://en.wikipedia.org/w/index.php?title=Oireachtas_na_Gaeilge&oldid=1296394643|journal=Wikipedia|language=en}}</ref> * Robert Louis Stevenson, ''An Fuadach'' (trans. of ''Kidnapped''), Leabhar Breac, 2016. * Clement Clarke Moore, ''Cuairt San Nioclás'' (trans. of ''A Visit from St. Nicholas'', or "'Twas the Night Before Christmas"), Leabhar Breac, 2022. '''''The Corto Maltese Graphic Novels''''' Written in Italian by Hugo Pratt and translated by Ó Scolaí, both adults and teenagers read this series of Italian adventure graphic novels.<ref>{{Cite journal|date=2025-07-01|title=Corto Maltese|url=https://en.wikipedia.org/w/index.php?title=Corto_Maltese&oldid=1298285365|journal=Wikipedia|language=en}}</ref> Ó Scolaí's '''translation of ''Corto Maltese''''' was listed in 2017 among "The 30 Irish books that Irish people love."<ref>{{Cite journal|last=Ó Murchú|first=Eoin P.|date=09/06/2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil [The 30 Irish books that Irish people love]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}</ref> * Hugo Pratt, ''Corto: Port na Farraige Goirt'', Leabhar Breac, 2013. * Hugo Pratt, ''Corto: The Golden House in Samarkand'', 2014. * Hugo Pratt, ''Corto: Na Liopard-Fhir ó Rufiji'' (trans. of ''Corto: The Leopard Men of Rufiji''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: In Ainm Dé Uilthrócairigh'' (trans. of ''Corto: In the Name of God All-Merciful''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: Tóraíocht Eile'' (trans. of ''Corto: Another Quest''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: Sa tSibéir'' (trans. of ''Corto: In Siberia''), Leabhar Breac, 2016. '''''Other Translations for Children''''' Ó Scolaí has translated into Irish six books from the ''Le Pavillon Noir'' (trans., ''Jolly Roger'') series by Alain Surget; four books from the ''Catalan First Steps'' series by Enric Lluch Girbés and the ''Caitlín & Cormac'' series by Joan Carles; three books from the ''Louisette le Taupe'' series by Bruno Heitz, and three books from the ''Loup'' series by Orianne Lallemand. === Plays and Screenplays === ==== Stage Plays ==== Ó Scolaí was writer and director of the original productions of two plays in the ''Trí Bhraon'' (trans., ''Three Drops'') trilogy; ''Coinneáil Orainn'' was directed by Darach Mac Con Iomaire and staged by An Taibhdhearc. All three plays have been published in book form by Leabhar Breac. * ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32553|title=Coinneáil Orainn|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}</ref> The first play in the ''Trí Bhraon'' (''Three Drops'') trilogy. [[wikipedia:Taibhdhearc_na_Gaillimhe|An Taibhdhearc]], the national Irish-language theatre of Ireland, toured the country in 2005 with ''Coinneáil Orainn''.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/coinneail-orainn/|title=Coinneáil Orainn|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Walter Macken Prize, 2005; BBC Stewart Parker Award, 2006.<ref>{{Cite web|url=https://irishplayography.com/person/darach-scola|title=Darach Ó Scolaí|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}</ref> * ''Branwen'', 2006, by Darach Ó Scolaí and Ifor ap Glyn, in Irish, Welsh and English, co-produced by Project Arts Centre and Llwyfan Gogledd Cymru, toured the Republic of Ireland and Wales.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32418|title=Branwen|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> * ''An Braon'' Aníos (trans., ''Rising Damp''), 2006, directed by Ó Scolaí.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32461|title=An Braon Aníos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> The second play in the ''Trí Bhraon'' (''Three Drops'') trilogy. “The Salamandar company toured the country in 2006-07 with this play, and Salamandar also produced a radio version of the play for RTÉ Raidió na Gaeltachta in 2009.”<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/an-braon-anios/|title=An Braon Aníos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Craos'' (trans., ''Gluttony''), 2008, directed by Ó Scolaí.<ref name=":13">{{Cite web|url=https://irishplayography.com/play?playid=32867|title=Craos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> The third play in the ''Trí Bhraon'' (''Three Drops'') trilogy, it toured to Cork and Belfast.<ref name=":13" /> A review of the 2008 Salamander performance in the ''Irish Times'' says, “a humorous play which offers plenty to think about, fine acting, and sparklingly witty dialogue.”<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/craos-2/|title=Craos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''A+E'', 2008, by Ríonach Ní Néill and Darach Ó Scolaí, "dance and music drama," co-produced by Ciotóg and Salamandar.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32962|title=A+E|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> * ''An tSeanbhróg'' (trans., ''The Old Shoe''), 2009, produced by Salamander<ref>{{Cite web|url=https://irishplayography.com/play?playid=33042|title=An tSeanbhróg|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> and staged in the Axis Arts Centre, Dublin, and the Letterkenny Arts Centre. * '''In ''Mhuir Fhíondorcha/The Wine-Dark Sea: The Homer Project'', Ó Scolaí's translation of Homer's Cyclops story, performed at the 2019 IMRAM festival'''.<ref>{{Cite news|url=https://www.irishtimes.com/culture/books/imram-a-festival-celebrating-the-irish-language-1.4047610|title=Imram: a festival celebrating the Irish language. Liam Carson reveals the myths and legends appearing in this year’s programme|last=Carson|first=Liam|date=11 October 2019|work=The Irish Times|access-date=15 October 2025}}</ref> ==== Screenplays ==== * ''Cosa Nite'' (trans., ''Washed Feet''), short film, 1998 (dir. Dearbhla Walsh, prod. Ciarán Ó Cofaigh, Rosg); "a prose version of ''Cosa Nite'' was published (Rosg 2000)."<ref name=":9" /> Nominated for an Irish Film and Television Award.<ref>{{Citation|title=Cosa Nite (Short 1998) - Awards - IMDb|url=https://www.imdb.com/title/tt0191917/awards/|accessdate=2025-08-25|language=en-US}}</ref> * ''Na Glúnta'' (trans., ''The Generations''), 2001<ref>{{Cite web|url=https://www.iftn.ie/production/production_companies/production_sub/feature/?act1=record&aid=70&rid=3917&tpl=filmography_dets&only=1&force=1|title=Na Glúnta {{!}} The Irish Film & Television Network|website=www.iftn.ie|access-date=2025-08-25}}</ref>, co-directors Ciarán Ó Cofaigh & Darach Ó Scolaí, prod. Ciarán Ó Cofaigh, Rosg. * ''An Leabhar'' (trans., ''The Book''), short film, 2000, (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg) Rosg, 2000.<ref>{{Citation|title=An Leabhar|url=https://www.imdb.com/title/tt0963767/|publisher=Bord Scannán na hÉireann / The Irish Film Board, ROSG|accessdate=2025-08-25|first=Robert|last=Quinn|others=Colm O&apos;Maonlai, Peadar O&apos;Treasaigh, Diarmuid Mac an Adhastair}}</ref> * ''Na Cloigne'' [trans., The Heads], 3-episide series, 2010 (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg), TG4.<ref>{{Cite web|url=https://www.imdb.com/title/tt1607924/|title=Na cloigne|date=2010|website=IMDb|access-date=25 August 2025}}</ref> === Nonfiction === Ó Scolaí's essays and lectures are published and his interviews are broadcast regularly, making for a large body of nonfiction critical and analytical work. Here are a few, almost all published in [https://comhar.ie/iris/scribhneoiri/darach-o-scolai/ Comhar]: * “Ceol Ciúin na nÉagmaise” (trans., “The Silent Music of Absence ['''the Fall?''']”), an essay on the 2014 Nobel Prize winner for literature, [[wikipedia:Patrick_Modiano|Patrick Modiano]], ''Comhar'', December 2014. * The Ó Cadhain Lecture: [https://leachtaiuichadhain.clo.ie/leachtai/2014 “Cuimhne agus Díchuimhne (trans., “Memory & Forgetfulness"]), 2014. * “Rithim agus Réim” ("Rhythm and Register"), a public lecture in the University College Dublin lecture series “Ó Thrácht go Twitter” (trans., "From Talk to Twitter"), 2014. * Review of Pádraig Ó Cíobháin’s ''Dréachta Chrích Fodla'', '''Comhar?, ??'''. * “Na Geilt i mBun an Tí” (trans., "The Madmen in Charge"), a talk at the Merriman Winter School, Comhar April 2012.<ref name=":18">{{Cite web|url=http://darachoscolai.ie/beathaisneis.html|title=Darach Ó Scolaí: Beathaisnéis|website=darachoscolai.ie|access-date=2025-09-26}}</ref> * The EFACIS podcast: Síle Ní Choincheannain talks to Darach Ó Scolaí about the historical novel. == Critical Reception == Ó Scolaí’s style has been called “crisp and elegant, and rich in language while being highly readable,”<ref>{{Cite journal|last=Heussaf|first=Anna|date=Summer 2025|title=Bláthnaid—A tale of love, violence and sorcery retold for readers today|url=https://booksirelandmagazine.com/blathnaid-a-tale-of-love-violence-and-sorcery/|journal=Books Ireland}}</ref> with “an unsurpassed richness and precision of language.”<ref name=":12">{{Cite journal|last=Ó Cróinín|first=Breandán|date=Summer 2025|title=unknown|journal=The Limerick Leader}}</ref> “Whimsical, hilarious, and subtly learned” is how Éilis Ní Dhuibhne described his writing.<ref name=":20" /> === Original Works === Ó Scolaí’s first novel, the 2007 ''An Cléireach'' (''The Clerk'') won two prizes and was described as “one of the great historical novels in the Irish language and among the best books written in the language since the beginning of this century.”<ref name=":12" /> Novelist Alan Titley says, “In ''An Cléireach'' Ó Scolaí creates the Ireland of war in the 17th century more fully than any other Irish writer on the subject of war since ''L’Attaque'' Eoghain Ó Thuairisc around 1798 [In ''An Cléireach'' cruthaíonn Ó Scolaí Éire an chogaidh san 17ú haois níos iomláine ná mar a dhein aon scríbhneoir Gaeilge eile ar ábhar cogaidh ó ''L’Attaque'' Eoghain Uí Thuairisc timpeall ar 1798].”<ref name=":11" />{{rp|25, Col. 1a}} Not all the reviews of this first novel were so positive, however; Proinsias O' Drisceoil says for the Irish Times says,<blockquote>This then is a novel in search of a plot, a story that attempts to attain a significance that eludes it.<ref>{{Cite news|url=https://www.irishtimes.com/news/a-disaffected-clerk-in-the-confederates-1.943070|title=A disaffected clerk in the confederates|last=O' Drisceoil|first=Proinsias|date=5 July 2008|work=The Irish Times|access-date=16 October 2025}}</ref></blockquote> In the ''Oxford Handbook of Modern Irish Fiction'' Pádraig Ó Siadhail analyzes rather than reviews ''An Cléireach'': <blockquote>In ''An Cléireach'', Ó Scolaí revisits the trauma of Cromwellian Ireland. The primary narrative device is once again the first-hand account, in this case by Tadhg Ó Dúbháin, a clerk and quartermaster in the Confederate Army in 1650. We sample the hardships, the friendships, the tensions, the rivalries, and the petty jealousies amongst comrades in arms, including remnants of the Gaelic literary class, as the Confederate soldiers, increasingly a rabble more than a cohesive unit, retreat in advance of Cromwell’s forces. ''An Cléireach'' concludes with the narrator and his family in exile in continental Europe. But along the retreat route, and central to the novel, members of the Confederate army camp, rest up, and tell versions of a story about the keeper of the treasured manuscript "Saltair an Easpaig" (The Bishop’s Psalter). Their versions raise issues about memory construction, the limitations of individual perspectives, personal agendas, and how minor changes in the telling of a story can alter our understanding of history, Thus, ''An Cléireach'' complements ''Fontenoy'' in moving beyond more realistic recreation of a historical event or period to interrogate the notion of history as construct.<ref>{{Cite book|title=The Oxford Handbook of Modern Irish Fiction|last=Ó Siadhail|first=Pádraig|publisher=Oxford University Press|year=2020|isbn=9780198754893|editor-last=Harte|editor-first=Liam|pages=598–99|chapter=Contemporary Irish Fiction}}</ref> </blockquote> Of ''Súil an Daill,'' in ''Nós'', Cathal Seoighe says, "The book deserves a significant place among the collection of high-quality books published in recent years that would make you feel sorry for someone who does not speak Irish [Tá áit shuntasach ag dul don leabhar i measc an chnuasaigh leabhair ar ardchaighdeán a foilsíodh le roinnt blianta anuas a d’fhágfadh trua agat don té atá gan Ghaeilge]."<ref>{{Cite journal|last=Seoighe|first=Cathal|date=09/26/2022|title=‘Dar leathmhagairle an diabhail, is leabhar den scoth é seo!’ ['According to the devil’s half-wit, this is a great book!’]|url=https://nos.ie/cultur/leabhair/dar-leathmhagairle-an-diabhail-is-leabhar-den-scoth-e-seo/|journal=Nós}}</ref> ''Bódléar'', Ó Scolaí's most recent book, is a “beautiful novel. There is magic and craftsmanship in it. A small miracle of a book and it is highly recommended.”<ref>{{Cite web|url=https://leabharbreac.com/bodlear-mioruilt-bheag-de-leabhar/|title=Bódléar: Míorúilt bheag de leabhar (Bódléar: A Small Miracle of a Book)|last=Ní Ghairbhí|first=Róisín|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Éilis Ní Dhuibhne in the ''Irish Times'' says,<blockquote>what a gem! An affectionately gentle satire of the Irish poetic scene during one creatively fluid 19th-century year, the story focuses on a Maigue poet and schoolteacher who goes on a trip to France and returns with camembert, a cafetiere, ‘Fleurs du Mal’, and a mission to convert the local traditionalists to la modernité. Whimsical, hilarious, and subtly learned, it’s absolutely delightful!<ref name=":20">{{Cite journal|last=Ní Dhuibhne|first=Éilis|date=30 June 2025|title=Éilís Ní Dhuibhne on the best Irish language books of 2025 so far: Including a history of the Gaeltacht Civil Rights Movements, a gem of a novel by Darach Ó Scolaí and Joe McHugh’s entertaining account of learning Irish|url=https://www.irishtimes.com/culture/books/review/2025/06/30/eilis-ni-dhuibhne-on-the-best-irish-language-books-of-2025-so-far/|journal=The Irish Times|pages=22}}</ref></blockquote> === Retellings and Translations === ==== ''Táin Bó Cuailnge'' ==== ''Táin Bó Cuailnge'' [''The Cattle Raid of Cooley''] is a modern edition of an 11th-century epic into modern Irish.<ref name=":3" /> Gearóid Denvir reviewed ''Táin Bó Cuailnge'' for ''Comhar'':<blockquote>Darach Ó Scolaí has ​​achieved a feat in this challenging reworking. He has found a high level of the Irish language to tell his story – as he has done before in his groundbreaking novel An Cléireach (2007, Leabhar Breac) and in his other prose works. This book is a decoration of the language, literature and culture of the Irish language, following the path of the old storytellers and writers and presenting material from the tradition to his own generation according to the understandings of his own time. The book will be a classic that will be of great interest to all readers of the Irish language, both ordinary readers, students, scholars and writers, and there should be a copy in every home in the country. [Tá éacht déanta ag Darach Ó Scolaí san athleagan dúshlánach seo. Tá réim ard den teanga Ghaeilge aimsithe aige lena scéal a inseacht – mar a rinne sé cheana ina úrscéal ceannródaíoch An Cléireach (2007, Leabhar Breac) agus i saothair eile phróis dá chuid. Is maisiú ar an teanga agus ar litríocht agus cultúr na Gaeilge an leabhar seo a leanas conair na seanscéalaithe agus na seanscríobhaithe agus ábhar de chuid an traidisiúin á chur i láthair a ghlúine féin aige de réir thuiscintí a linne féin. Clasaic a bheas sa leabhar a gcuirfidh léitheoirí uilig na Gaeilge, idir ghnáthléitheoirí, mhic léinn, scoláirí agus scríbhneoirí spéis thar na bearta ann, agus ba cheart cóip a bheith i chuile theach sa tír.]<ref name=":15">{{Cite journal|last=Denvir|first=Gearóid|date=April 2018|title=Táin Bó Cuailgne|url=https://comhar.ie/iris/78/4/leirmheas/|journal=Comhar|via=JSTOR}}</ref> </blockquote>Cathal Poirtéir says, "The freshness and richness of Ó Scolaí’s version are a joy …. The author delights us with the linguistic and stylistic richness of the ancient epic in a modern-Irish version that reflects the original’s spirit and language."<ref>{{Cite journal|last=Poirtéir|first=Cathal|date=May/June 2018|title=Leabhair Idir Lámha|url=https://www.jstor.org/stable/26564180|journal=Books Ireland|pages=46–47|via=JSTOR}}</ref>{{rp|47}} Novelist and academic Alan Titley calls Ó Scolaí's "a wonderful gutsy telling" of ''Táin Bó Cuailnge''.<ref>{{Cite news|url=https://www.irishtimes.com/culture/2023/03/11/the-tain-retold-maeve-and-ailills-spat-could-be-out-of-a-soap-opera/|title=The Táin retold: ‘Maeve and Ailill’s spat could be out of a soap opera’|last=Titley|first=Alan|date=11 March 2023|work=The Irish Times|access-date=16 October 2025}}</ref> ==== ''Deirdre'' ==== Marie Whelton, in "Léann Teanga" ("Language Studies"), in the 2024 ''An Reiviú'' says,<blockquote>this version [of ''Deirdre''] by Darach Ó Scolaí succeeds in skillfully capturing and portraying the complexity of gender and power issues in the ‘Deirdre’ tradition [éiríonn leis an leagan seo le Darach Ó Scolaí castacht cheisteanna na hinscne agus na cumhachta i dtraidisiún scéal Dheirdre a ghabháil agus a léiriú go sciliúil]. … There is no doubt that this new version greatly contributes to the legacy of the story and that it revives that legacy thoughtfully and artistically [Níl amhras faoi ach go gcuireann an leagan úr seo go mór le hoidhreacht an scéil agus go ndéanann sé an oidhreacht sin a athbheochan go tuisceanach agus go healaíonta.].<ref name=":16">{{Cite web|url=https://www.tara.tcd.ie/tara8/server/api/core/bitstreams/3c20175a-7631-44b2-8b0f-f454edd712b4/content|title=An Artistic Retelling of Deirdre's Tale and the Defeat of Conor Review of Deirdre or the Ship of Mac Uisnigh by Darach Ó Scolaí [Athinsint Ealaíonta ar Oidhe Dheirdre agus ar Ansmacht Chonchúir Léirmheas ar Deirdre nó Loingeas Mhac Uisnigh le Darach Ó Scolaí]|last=Whelton|first=Marie|date=2024|website=The Review [An Reiviú], Language Studies [Léann Teanga]|access-date=25 September 2025}}</ref></blockquote> === Works for Young Readers === Meadhbh Ní Eadhra said of ''Bodach an Chóta Lachna'' that it was "Beautiful Irish, but easy to understand for young readers."<ref>Ní Eadhra, Meadhbh. In ''Gaelscéal'', qtd. in "Bodach an Chóta Lachna" https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/.</ref> == Awards and Honors == Ó Scolaí's works are regularly nominated and make the short list for prizes, an honor in itself, but they are generally not listed here unless they are named as the first-place winner in their category. === Oireachtas Prize === The Oireachtas Prize is the literary prize awarded by [[wikipedia:Oireachtas_na_Gaeilge|Oireachtas na Gaeilge]], the annual arts festival dedicated to Irish language, arts and culture. Darach Ó Scolaí has won the Oireachtas Prize for Literary Fiction three times, once for ''An Cléireach'' (''The Clerk'') in 2007, for ''Súil an Daill'' (''The Eye of the Blind'') in 2021 and for ''Bódléar'' in 2024. * 2007, for ''An Cléireach'' (trans., ''The Clerk'') — “(a special prize commemorating the 400th anniversary of the foundation of Coláiste na nGael in Louvain, awarded under the auspices of the Franciscan Province of Ireland). The prize of €10,000 was the largest prize ever awarded to an Irish language novel [(duais speisialta chomórtha 400 bliain bhunú Choláiste na nGael i Lobháin a bronnadh faoi urraíocht Phroibhinse Phroinsiasach na hÉireann). Ba é an duais €10,000 sin an duais ba mhó a bronnadh riamh ar úrscéal Gaeilge].”<ref name=":18" /> * 2021, for ''Súil an Daill'' (''The Eye of the Blind'') * 2024, for ''Bódléar'' === Ó Shúilleabháin Award, Irish language “Book of the Year” === The first prize of this award includes €5,000 to the publisher and €2,500 to the author of the winning work.<ref name=":10">{{Cite journal|date=15 August 2023|title=20 saothar san iomaíocht do ‘Leabhair Ghaeilge na Bliana 2023’|url=https://tuairisc.ie/20-saothar-san-iomaiocht-do-leabhair-ghaeilge-na-bliana-2023/|journal=Tuairisc}}</ref> * ''An Cléireach'' (''The Clerk'').<ref>{{Cite web|url=http:/www.gaelport.com/uploads/documents/edition19.html|title=Eagrán / Edition 19 - 04 11 2008|date=4/11/2008|website=Internet Archive|archive-url=https://web.archive.org/web/20130525011340/http:/www.gaelport.com/uploads/documents/edition19.html|archive-date=25 May 2013|access-date=25 August 2025}}</ref> * ''Táin Bó Cuailnge'', 2018. * ''Bódléar'', 2025. ==== De Bhaldraithe Award ==== The Gradam de Bhaldraithe is awarded to the best work in translation.<ref name=":10" /> * ''Cuairt San Nioclás,'' a translation of Clement Clarke Moore's ''A Visit from St. Nicholas'', or "'Twas the Night Before Christmas."<ref name=":10" /> ==== Other ==== * Walter Macken Prize, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005 * Bháiteir Uí Mhaicín Memorial Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005<ref>{{Cite news|url=https://www.irishtimes.com/gaeilge/tuarascail/duais-oireachtais-1.501571|title=Oireachtas Prize: Over €50,000 was awarded to writers in the Oireachtas Literary Competitions at an event in Dublin last night. Winners… [Duais Oireachtais: Bronnadh breis agus €50,000 ar scríbhneoirí i gComórtais Liteartha an Oireachtais ar ócáid i mBaile Átha Cliath aréir. Bhuaigh…]|work=5 October 2005|access-date=15 October 2025}}</ref> * BBC Stewart Parker Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2006 * The Aodán Mac Póilín Commemorative Prize, for ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), 2017 == External Links == * Leabhar Breac website: https://leabharbreac.com/en/ * Leabhar Breac Facebook pages: * Rosg website: [http://www.rosg.ie/en/ <nowiki>http://ww</nowiki>w.rosg.ie/en/] * Art on the Island (Ealaín ar Oileán) website, archived at the Wayback Machine: https://web.archive.org/web/20130601000520/http://ealainaroilean.ie/ 31 March 2012, 1 June 2013 and 8 January 2014 * Darach Ó Scolaí's website Archived 25 September 2015 at the Wayback Machine: https://web.archive.org/web/20150925103456/http://darachoscolai.ie/ * Youtube video of [https://www.youtube.com/watch?v=OlP2AmSBzXc Breandán Ó Cróinin introducing Deirdre at the book launch] in the pub Tigh Mholly (Molly’s House). == Primordial Ooze == * Known for his sensitivity to language and voices. * Finish scanning through JSTOR * Scan through Irish Times, 56 hits * Check Goodreads * Check YouTube (In the spring of 2013, the arts programme Imeall interviewed the author on TG4.) * Check both Wikipedias for pages on the origins of the retold tales (like Deirdre) and link to this article * Propose link from University of Galway page once Darach’s is up * Write Irish National Biography (<nowiki>https://www.dib.ie</nowiki>) to propose an article about Darach once the Wikip article is done? See what they say. * Link to Ó Scolaí from the Wikipedia * Make sure links '''to''' Wikipedia in the actual encyclopedia work right === Not Placed Yet === * "So here are the books that Irish people love the most! [Mar sin seo iad na leabhair is gile leis na Gaeil!]" — "32. An Cléireach – Darach Ó Scolaí (2)" [18 books got 2 votes, and then they're alphabetized by author's last name, so the 32 of 34 doesn't signify the specificity it seems to]<ref name=":14">{{Cite journal|last=Ó Murchú|first=Eoin P.|date=9 June 2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil. [The 30 best Irish books for Irish people]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}</ref> * "Below is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers.Here is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers [Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí.Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí]." "Corto Maltese – Hugo Pratt (aistrithe ag Darach Ó Scolaí)"<ref name=":14" /> * "Ceann eile de bhuaicphointí na hÉigse a bheidh sa seisiún le Darach Ó Scolaí, duine d’úrscéalaithe móra na Gaeilge, agus duine de chomhbhunaitheoirí teach foilsitheoireachta Leabhar Breac. [Another highlight of the Éigse will be the session with Darach Ó Scolaí, one of the great Irish language novelists, and one of the co-founders of the publishing house Leabhar Breac.]"<ref name=":17">{{Cite journal|last=Nós|date=4 May 2023|title=Éigse na Bruiséile le filleadh i mí na Bealtaine. [Éigse na Bruséile to return in May]|url=https://nos.ie/cultur/eigse-na-bruiseile-le-filleadh-i-mi-na-bealtaine/|journal=Nós}}</ref> === Things Taken Out for Now === “’The play is a comedy about language, lies, bureaucracy and Gaeltacht grants, in the tradition of Myles na Gcopaleen,’ according to Norma-Jean Kenny in the ''Galway Advertizer'', ‘in which the author comments and criticizes the institutions of the Irish language in Ireland without ceasing.’" Supposedly a quotation by Gearóid Denvir reviewing ''Táin Bó Cuailnge'' for ''Comhar'' (but I don't find it in the article): This book has long been needed by Irish language readers and there is no doubt that it will become a classic in time and surpass Thomas Kinsella’s English version. This version remains faithful to the language of the original while at the same time finding an appropriate language in today’s Irish. Ó Scolaí masterfully overcomes the difficulties of the original’s rhetorical difficulties and the versions of the original poetic texts are extremely effective.[supposedly <ref name=":15" />] “The biggest prize ever awarded for a novel in Irish was presented at a special ceremony in the National Concert Hall in Dublin, today (Thursday, 4 October 2007). Darach Ó Scolaí, writer, artist & playwright from Casla, Co. Galway, was awarded €10,000 for his literary novel, ‘An Ardscoil’. This work, under the new title ‘An Cléireach’, will be launched at Oireachtas na Samhna in Westport in November. This is the first novel from his pen, a story set in the late seventeenth century. This competition was sponsored by the Franciscan Province of Ireland.” (archive, Oireachtas na Gaeilge site, 04 October, 2007) ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. number 2 in ''Comhar'' literary magazine’s list of best books of 2021. '''{6}.''' *William Shakespeare, ''Romeo agus Juliet'' (trans. of ''Romeo and Juliet''), Leabhar Breac, 2016. *Jonathan Swift, ''Camchuairt Ghuilivéir'' (trans. of ''Gulliver's Travels''), Leabhar Breac, 2016. *Hugo Pratt, ''Corto Maltese'' '''''Flag of Bones (Bratach na gCnámh) Series''''' Leabhar Breac published the Bratach na gCnámh series of books for young readers. Written in French by Alain Surget, illustrated by Annette Marnat and translated by Darach Ó Scolaí, this series uses the history of Caribbean Sea pirates<ref>{{Cite web|url=https://leabharbreac.com/en/product-category/alain-surget/|title=Alain Surget Archives|website=Leabhar Breac|language=en-US|access-date=2025-09-30}}</ref>: *Alain Surget, ''Éalú as Páras'' (''Escape from Paris''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''Oilean na Siorcanna'' (''Shark Island''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''Long na dTaibhsi'' (''Ship of the Ghosts''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''San Ochtapas Dubh'' (''In the Black Octopus''), Annette Marnat (Illustr.), Leabhar Breac, 2013. * Alain Surget, ''San Ionsai ar Veracruz'' (''The Attack on Veracruz''), Annette Marnat (Illustr.), Leabhar Breac, 2013. '''''For "First Readers" (children to 6 years old or so)''''' These books were written originally in Catalan by Spanish author Enric Lluch Girbés and translated into Irish by Ó ScolaÍ: *Enric Lluch, ''Ag Péinteáil an Tí'' (''Painting the House''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''An Colúr Bacach'' (''The Lazy Dove''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''An Phluais'' (''The Cave''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''Madra Dhaideo'' (''Grandpa's Dog''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''Fiacail Mháire'' (''Mary's Tooth''), Anna Clariana (Illustr.), Leabhar Breac, 2017. '''''Bruno Heitz''''' Leabhar Breac published a series of 3 Heitz books for small children. Published originally in French, this series of three comic books is about a blind mole named Cáitín Chaoch in Irish (and ''Louisette la taupe'' in French).<ref>{{Cite web|url=https://leabharbreac.com/en/product-category/bruno-heitz-en/|title=Bruno Heitz Archives|website=Leabhar Breac|language=en-US|access-date=2025-10-02}}</ref> Ó Scolaí translated these: *Bruno Heitz (author and illustr.), ''Práinneach'' (''Urgent''), Leabhar Breac, 2020 *Bruno Heitz (author and illustr.), ''Preab san Aer'' (''Bounce in the Air''), Leabhar Breac, 2020. '''''Books for Toddlers''''' Leabhar Breac has published 14 books written by French author Orianne Lallemand's and illustrated by Eleonore Thuillier, about Lallemmand's popular character Loup, Wolf. These are translated by Ó Scolaí: *Orianne Lallemand, ''An Mac Tire a Raibh Faitios an Domhain Air'' (trans. of ''The Son Who Saw the World in His Eyes''), Eleonore Thuillier  Illustr.), Leabhar Breac, 2018. *Orianne Lallemand, ''Macan agus an Goban'' (trans. of ''Macan and the Goblin''), Eleonore Thuillier (Illustr.), Leabhar Breac, 2018. * Orianne Lallemand, ''A Mac Tíre a Chuaigh go Tóin na Farraige'' (trans. of ''The Wolf Who Went to the Bottom of the Sea''), Éléanore Thuillier  (Illustr.), Leabhar Breac, 2019. '''''Board Books (for babies)''''' J. C. (Joan Carles) Girbés Aparisi is a Catalan author and editor. These books were written in Catalan and translated by Ó Scolai. *J. C. Girbés, ''An Phicnic'' (''The Picnic''), Silvia Ortega (Illustr.), Leabhar Breac, 2013. * J. C. Girbés, ''An Chóisir'' (''The Party''), Silvia Ortega (Illustr.), Leabhar Breac, 2013. *J. C. Girbés, ''Lá Mór Fada'' (''A Long Day''), Silvia Ortega (Illustr.), Leabhar Breac, 2014. *J. C. Girbés, ''Tabhair Leat do Leabhar'' (''Bring Your Book''), Silvia Ortega (Illustr.), Leabhar Breac, 2014. ==== Gradam Réics Carló ==== The Réics Carló prize is awarded for the best book in the Irish language for young readers. It is named for one of the characters of 20th-century writer [[wikipedia:Cathal_Ó_Sándair|Cathal Ó Sándair (Charles Saunders)]]. * ''An Bradán Feasa'' was “shortlisted for the Réics Carlo award 2010.”<ref name=":9" /> == References == {{reflist}} fa8oweof59meh4c3u7j6vatf8jijb3f 0 268894 2844130 2779911 2026-11-03T08:09:50Z James500 297601 /* Japan */ Add 2844130 wikitext text/x-wiki {{Bibliography}} See also [[Universal Bibliography#History and Geography|History and Geography]]. This part of the [[Universal Bibliography]] is a bibliography of history. See [[w:Bibliography of encyclopedias: history]], [[w:List of archaeology and history books]] and [[w:Category:Works about history]]. ==Metabibliography== See [[w:Category:Bibliographies of history]] *D'Aniello (ed). Teaching Bibliographic Skills in History. 1993 *Besterman. History and Geography: A Bibliography of Bibliographies. 1972 [https://books.google.co.uk/books?id=RzA5AAAAIAAJ] *Guide to Periodicals and Bibliographies Dealing with Geography, Archaeology, and History [https://books.google.co.uk/books?id=V6tCAAAAIAAJ] *A Consumer's Guide to Research Guides for Historical Literature. American Library Association. 1990 [https://books.google.co.uk/books?id=HkvgAAAAMAAJ] *The American Historical Association's Guide to Historical Literature. 1995 [https://books.google.co.uk/books?id=lU3gAAAAMAAJ] *George Matthew Dutcher. A Guide to Historical Literature. P Smith. 1949. [https://books.google.co.uk/books?id=ZEbgAAAAMAAJ Google Books] *Adams. A Manual of Historical Literature. 1882 [https://archive.org/details/manualofhistoric00adamuoft], 3rd Ed: 1889 [https://archive.org/details/manualofhistoric00adam] [https://archive.org/details/manualofhistoric00adamrich] *Historical Abstracts: Bibliography of the World's Historical Literature. [https://books.google.co.uk/books?id=CHFsAAAAIAAJ] *International Bibliography of Historical Sciences. De Gruyter *Sonnenschein. A Bibliography of History and Historical Biography. 1897 [https://archive.org/details/abibliographyhi00unkngoog] *Andrews, Gambill and Tall. Bibliography of History for Schools and Libraries. 1910 [https://archive.org/details/bibliographyofhi00andruoft] [https://archive.org/details/bibliographyofhi00andr] [https://archive.org/details/abibliographyhi00tallgoog] *A Bibliography of History: Being a Collection of Reading Lists from Courses in History and Allied Fields. Harvard Graduate History Club. 1953 [https://books.google.co.uk/books?id=klZTAAAAYAAJ] *Grose (ed). A Select Bibliography of History. 1970 [https://books.google.co.uk/books?id=Af1CAAAAIAAJ] *Fritze, Coutts and Vyhnanek. Reference Sources in History: An Introductory Guide. 1990 [https://books.google.co.uk/books?id=Fk7gAAAAMAAJ]. 2nd Ed: 2004 [https://books.google.co.uk/books?id=7VyOANl2qxoC&pg=PP1#v=onepage&q&f=false] *A Consumer's Guide to Ready Reference Sources in History. American Library Association. 1990 [https://books.google.co.uk/books?id=n03gAAAAMAAJ] *Boehm and Birkos. Reference Works: History and Related Fields. 1967 [https://books.google.co.uk/books?id=W0hBAAAAIAAJ] *Bibiographies and Indexes in World History. ISSN 0742-6852. [https://books.google.co.uk/books?id=NiHgAAAAMAAJ Issue 12] *Roach (ed). A Bibliography of Modern History. CUP. 1968 [https://books.google.co.uk/books?id=hhU9AAAAIAAJ&pg=PR3#v=onepage&q&f=false] *Medlicott. Modern European History, 1789-1945: A Select Bibliography. 1960. Revised: 1973. [https://books.google.co.uk/books?id=qzEsAQAAIAAJ] *Rouse, Claxton and Metzger. Serial Biblographies for Medieval Studies. University of California Press. 1969 [https://books.google.co.uk/books?id=OyJj6gGgYZgC&pg=PR3#v=onepage&q&f=false] *International Medieval Bibliography. University of Leeds *Davis. Medieval European History 395-1500: A Select Bibliography. Historical Association. 2nd Ed: 1968 [https://books.google.co.uk/books?id=kH4LAAAAMAAJ] *Iliff. A Guide to the Literature in English on the Teaching of History. University of California. 1914. [https://books.google.co.uk/books?id=MmpIAQAAMAAJ] Historiography: *Moyer. Historiography: Oxford Bibliographies Online Research Guide. OUP. 2010. [https://books.google.co.uk/books?id=4BfBzJTAO_0C&pg=PP1#v=onepage&q&f=false] Military and Naval: *Bibliographies of Battles and Leaders. [https://books.google.co.uk/books?id=i-dmAAAAMAAJ Issue 17] *DeVries. A Cumulative Bibliography of Medieval Military History and Technology. 2002. Update 2003-2006, published 2008. [https://books.google.co.uk/books?id=84oTXprXf0MC&lpg=PP1&pg=PP1#v=onepage&q&f=false] *Horward. Napoleonic Military History: A Bibliography. Garland. 1986. [https://books.google.co.uk/books?id=6YAWAQAAIAAJ] *Schneid. Napoleonic Wars. The Essential Bibliography Series. 2012 [https://books.google.co.uk/books?id=n89BDcWXFpIC&pg=PP1#v=onepage&q&f=false] *Whaley. The Impact of Napoleon, 1800-1815: An Annotated Bibliography. 1997. [https://books.google.co.uk/books?id=NIjgAAAAMAAJ] *Bruce, Anthony. A Bibliography of British Military History: From the Roman Invasions to the Restoration, 1660. K G Saur. 1981. [https://books.google.co.uk/books?id=w59sDwAAQBAJ&pg=PR1#v=onepage&q&f=false] *Rasor. English/British Naval History to 1815. Praeger. 2004. The Seaforth Bibliography. Pen and Sword. 2008. [https://books.google.co.uk/books?id=KSDAAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Rasor. The Spanish Armada of 1588: Historiography and Annotated Bibliography. 1993. [https://books.google.co.uk/books?id=P1JnAAAAMAAJ] *Lynch. The War at Sea: France and the American Revolution: A Bibliography. [https://books.google.co.uk/books?id=QSi_pS_tgSoC&pg=PP5#v=onepage&q&f=false] Science: *Whitrow. ISIS Cumulative Bibliography: A Bibliography of the History of Science Formed from ISIS Critical Bibliographies 1-90, 1913-65. [https://books.google.co.uk/books?id=9rEQAQAAMAAJ] *Wehefritz. Bibliography on the History of Chemistry and Chemical Technology 17th to the 19th Century. 1994. [https://books.google.co.uk/books?id=j6fXUe6DO7EC&pg=PP2#v=onepage&q&f=false] *Bibliography of the History of Medicine. (National Library of Medicine). [https://books.google.co.uk/books?id=qS-ZXvSFVYgCpg=PR1#v=onepage&q&f=false] Philosophy: *Jasenas. A History of the Bibliography of Philosophy. 1973. [https://books.google.co.uk/books?id=80erVxQc7qoC&lpg=PA99&pg=PP1#v=onepage&q&f=false] Religion: *International Bibliography of the History of Religions. [https://books.google.co.uk/books?id=cK83AAAAIAAJ&pg=PP1#v=onepage&q&f=false 1964 volume], 1969. [https://books.google.co.uk/books?id=IQ0VAAAAIAAJ&pg=PP1#v=onepage&q&f=false 1973 volume], 1979. *Chaniotis. Greek Religion. (Oxford Bibliographies Online Research Guide). 2010. [https://books.google.co.uk/books?id=jUmtObc-ECkC&pg=PP1#v=onepage&q&f=false] *Taneti. History of the Telugu Christians: A Bibliography. (ATLA Bibliography Series, number 60). 2011. [https://books.google.co.uk/books?id=W1xAHHzUbNsC&pg=PP1#v=onepage&q&f=false] *Norton. Religious Newspapers in the Old Northwest to 1861: A History, Bibliography, and Record of Opinion. 1977. [https://books.google.co.uk/books?id=_BUPAAAAMAAJ] Classics: *Calder and Kramer. An Introductory Bibliography to the History of Classical Scholarship: Chiefly in the XIXth and XXth Centuries. 1992. *Calder and Smith. A Supplementary Bibliography to the History of Classical Scholarship: Chiefly in the XIXth and XXth Centuries. 2000. [https://books.google.co.uk/books?id=pFZEDkIe33MC&pg=PP1#v=onepage&q&f=false] *Hansen. A Bibliography of Danish Contributions to Classical Scholarship from the Sixteenth Century to 1970. 1977. [https://books.google.co.uk/books?id=v3cNAQAAIAAJ] *Andersen. Danish Contributions to Classical Scholarship, 1971-1991: A Bibliography. 2004. [https://books.google.co.uk/books?id=S2O2rqwdWIMC&pg=PP1#v=onepage&q&f=false] North America: *Bibliographic Guide to North American History. G K Hall. 1979. 1980. 1982. 1985. 1986. [https://books.google.co.uk/books?id=IKbfAAAAMAAJ] *Bibliographies and Indexes in American History [https://books.google.co.uk/books?id=m46OAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Brasseaux and Conrad. A Bibliography of Scholarly Literature on Colonial Louisiana and New France. 1992. [https://books.google.co.uk/books?id=zk8WAQAAIAAJ] British Isles *[[w:Bibliography of British and Irish History|Bibliography of British and Irish History]] *Annual Bibliography of British and Irish History. Royal Historical Society. *Ireland, Stanley. Roman Britain: A Sourcebook. Croom Helm. 1986. [https://books.google.co.uk/books?id=obkOAAAAQAAJ] British Empire, Colonies and Commonwealth *Porter, Andrew. Bibliography of Imperial, Colonial, and Commonwealth History Since 1600. Oxford University Press. 2002. [https://books.google.co.uk/books?id=4r7fAAAAMAAJ] France: *McCrank. Bibliographical Foundations of French Historical Studies. Haworth. 1992. Routledge 1996. [https://books.google.co.uk/books?id=uydNngEACAAJ] [https://books.google.co.uk/books?id=YgPTwAEACAAJ] *Bowditch, Grew, Geiger. A Selected Bibliography on Modern French History, 1600 to the Present. 1974. [https://books.google.co.uk/books?id=2ngWAQAAMAAJ] *Echard. Foreign policy of the French Second Empire: A Bibliography. 1988. (Bibiographies and Indexes in World History, issue 12) *Young. French Foreign Policy 1918-1945: A Guide to Research and Research Materials. 1981. Revised: 1991. [https://books.google.co.uk/books?id=1Ph3s4sJvcgC&pg=PR2#v=onepage&q&f=false] Scandinavia: *Munch-Petersen (ed). Guide to Nordic Bibliography. 1984. (016:9 History, 016:902 Archaeology, 016:908 Local history). Pages [https://books.google.co.uk/books?id=Xh5YCrl2oGoC&pg=PA194#v=onepage&q&f=false 194] to 206. Norway: *Bibliografi Til Norges Historie *Hill: Norwegian Local History: A Bibliography of Material in the Collections of the Memorial Library, University of Wisconsin-Madison. 1989. [https://books.google.co.uk/books?id=n0VEAAAAMAAJ] Denmark: *Dansk Historisk Bibliografi *Jansen and Riis. A Select Bibliography of Danish Works on the History of Towns Published 1960-1972. 1973. [https://books.google.co.uk/books?id=e9knAQAAIAAJ] *Tamm. The History of Danish Law: Selected Articles and Bibliography. 2011. [https://books.google.co.uk/books?id=oT4qKQEACAAJ] *Snorrason. The History of Medicine in and about Denmark: Sources and Bibliography. 1968. Visigoths: *Ferreiro. The Visigoths in Gaul and Spain: A.D. 418-711: A Bibliography. 1988. [https://books.google.co.uk/books?id=lOZ8DR6w-68C&pg=PP1#v=onepage&q&f=false] United States: *Bibliographies of States of the United States. [https://books.google.co.uk/books?id=VBVNm8Zgtn8C&pg=PP1#v=onepage&q&f=false No 8] *Flagg and Jennings. Bibliography of New York Colonial History. (New York State Library, Bulletin 56, Bibliography 24, February 1901) *Dickinson and Hitchcock. A Bibliography of Tennessee History, 1973-1996. 1999. [https://books.google.co.uk/books?id=jbc4BFCMEd8C&lpg=PR12&dq=Bibliography%20history&pg=PP1#v=onepage&q&f=false] *Schlinkert. Subject Bibliography of Wisconsin History. 1947. [https://books.google.co.uk/books?id=RRI2AQAAIAAJ] *Bradley. A Bibliography of Wisconsin's Participation in the War between the States. 1911. [https://books.google.co.uk/books?id=3QEuAAAAYAAJ] China *Wilkinson. [[w:Chinese History: A New Manual|Chinese History: A New Manual]] ==General Series== *Oxford Histories *Oxford Historical Monographs. (Oxford Monographs Series) *Cambridge Histories *Exeter Studies in History *Access to A Level History. Edward Arnold. *Seminar Studies in History. Longman. *Flagship History *Sutton Pocket Histories *Studies in Economic History. Macmillan. *Langer (ed). The Rise of Modern Europe. Harper. *New Oxford History of England *Commager and Morris (eds). New American Nation. Harper. *Brookeman and Issel (eds). The Contemporary United States. Macmillan. *Perrot and Chipiez's Works on Ancient Art. [[s:Page:History of Art in Phrygia, Lydia, Caria and Lycia.djvu/423|Wikisource]] *Columbia Histories of Modern American Life. *Shotwell (ed). Records of Civilazation: Sources and Studies. Columbia University Press. [[s:Page:History of the Franks.djvu/319|(wikisource)]] ==General== *Gardiner, Juliet (ed). What is History Today? Macmillan Education. 1988 *Cambridge Ancient History *Cambridge Medieval History *Cambridge Modern History *Durant, Will and Ariel. The Story of Civilization. General television series *[[w:Timewatch|Timewatch]]. 1982 onwards. BBC2. [Television series] *[[w:Secret History (TV series)|Secret History]]. 1991 onwards. Channel 4. [Television series] ==World== *Wetterau, Bruce (compiler). Concise Dictionary of World History. Macmillan. 1983. Robert Hale. 1984. *Roberts, J M. Shorter Illustrated History of the World. Helicon Publishing. 1993. BCA. 1994. ==Millennia, centuries and decades== For chronologies of particular millennia and centuries, see Chronology 2nd millennium AD *[[w:CNN Millennium|CNN Millennium]]. 1999. [Television series] [About the 2nd millennium AD] 20th century *[[w:People's Century|People's Century]]. 1995 to 1997. [Television series] Decades *Series: **Decades in Global History: The Global 1920s [https://books.google.co.uk/books?id=jSp-CwAAQBAJ&pg=PP1#v=onepage&q&f=false], 1930s [https://books.google.co.uk/books?id=-gkqDwAAQBAJ&pg=PP1#v=onepage&q&f=false], 1960s [https://books.google.co.uk/books?id=FMItDwAAQBAJ&pg=PA3#v=onepage&q&f=false], 1970s [https://books.google.co.uk/books?id=eQtpDwAAQBAJ&pg=PA2#v=onepage&q&f=false], 1980s [https://books.google.co.uk/books?id=cnGPDwAAQBAJ&pg=PA1#v=onepage&q&f=false] ==Europe== *Thomson, David. Europe since Napoleon. Longmans. 1957. 2nd Ed: 1962. ==Britain== "Roman Britain", The Times, series of articles in 1996 *Alan Hamilton, "Invaders who left a cultural revolution in their wake", The Times, 3 August 1996, p 8 *Alan Hamilton, "Impressing the barbarians along the wild frontier", The Times, 5 August 1996, p 6 **Emma Wilkins, "Keeping up appearances was time well spent for garrison town women", The Times, 5 August 1996, p 6 **"How to see the best of the sites", The Times, 5 August 1996, p 6 **"Cives Romani Sumus", The Times, 5 August 1996, p 17 *The Times, 6 August 1996 *Alan Hamilton, "Unruly Tykes stretched Ninth Legion to the limit", The Times, 7 August 1996, p 7 **"Where all roads lead to Romans", The Times, 7 August 1996, p 7 *Alan Hamilton, "Port city that leaves scholars with two unsolved questions", The Times, 8 August 1996, p 9 **"Bath-house lies buried under Spud-U-Like", The Times, 8 August 1996, p 9 *Bill Frost, "Legions' elephants swept all before them at Colchester", The Times, 9 August 1996, p 9 **John Young, "Invasion troops took over native spa for rest and recreation", The Times, 9 August 1996, p 9 *Alan Hamilton, "Londinium's treasures lie beneath every City street", The Times, 10 August 1996, p 8 *Bill Frost, "Families buried their riches before they fled", The Times, 10 August 1996, p 8 *Alan Hamilton, "Vanishing remains of an emperor's highland fling", The Times, 12 August 1996, p 6 **Alan Hamilton, "Tracing footsteps of the legions", The Times, 12 August 1996, p 6 *Bill Frost, "Ideal home that fell apart when the peasants moved in", The Times, 13 August 1996, p 7 **Bill Frost, "Latin ghosts haunt town preserved by centuries of neglect", The Times, 13 August 1996, p 7 **Emma Wilkins, "Fourth largest city grew from border fort built to quell 'little Britons'", The Times, 13 August 1996, p 7 *Bill Frost, "The Battle of Britain, Roman style", The Times, 14 August 1996, p 6 *Emma Wilkins, "Sumptuous reward for a co-operative chieftan", The Times, 15 August 1996, p 10 *Alan Hamilton, "The fortress that was left to its own peaceful resources", The Times, 16 August 1996, p 8 **"Archway survives ravages of time", The Times, 16 August 1996, p 8 **"Beaten Iceni were herded into Norfolk new town", The Times, 16 August 1996, p 8 *Emma Wilkins, "Empire's home-builders were liberated women", The Times, 17 August 1996, p 8 **Alan Hamilton, "Gourmet invader added spice to Celtic life", The Times, 17 August 1996, p 8 **John Young, "Frontier town that embraced civilisation", The Times, 17 August 1996, p 8 *John Young, "Welsh fortress where troops faced tribes", The Times, 19 August 1996, p 7 **"Roman Britain's secrets explored", The Times, 19 August 1996, p 19 ==Japan== *David Murray. Japan. (The Story of the Nations). T Fisher Unwin. Rev Ed: 1920: [https://books.google.co.uk/books?id=x3HUAAAAMAAJ&pg=PR1#v=onepage&q&f=false]. *Walter Dickson. Japan. (Collier's Nations of the World, vol 29). 1898. [https://books.google.co.uk/books?id=Xpk-AAAAYAAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=Yr5IAQAAIAAJ&pg=PA3#v=onepage&q&f=false] Newspaper reprints *[https://www.japantimes.co.jp/tag/japan-times-gone-by/?type=column Japan Times Gone By]. The Japan Times. ==Records== Inns of court *[[s:The Records of the Honorable Society of Lincoln's Inn|The Records of the Honorable Society of Lincoln's Inn]]. Google Books: editions:44paS1hFG1MC ==Buildings and sites== *[[w:One Foot in the Past|One Foot in the Past]]. BBC2. 1993 to 2000. [Television series] *Branigan. [https://amp.theguardian.com/world/2009/dec/14/china-historic-sites-survey "China loses thousands of historic sites"]. The Guardian. 14 December 2009. *[https://amp.scmp.com/lifestyle/travel-leisure/article/3110351/historic-buildings-india-photographed-and-their-stories "Historical buildings in India photographed and their stories preserved before they succumb to decay and neglect"]. South China Morning Post. *Vijay. [https://m.economictimes.com/opinion/et-commentary/asi-and-nma-need-a-new-avatar-to-conserve-and-protect-indias-monuments/articleshow/100021418.cms "ASI and NMA need a new avatar to conserve and protect India's monuments"]. The Economic Times. 5 May 2023. ==Ships== *International Register of Historic Ships ==Wars== *Purnell's History of the World Wars **Purnell's History of the First World War **Barrie Pitt (ed). [[w:Purnell's History of the Second World War|Purnell's History of the Second World War]]. Purnell's Illustrated History of the Second World War. *Ballantine's Illustrated History of the Violent Century **The Pan/Ballantine Illustrated History of World War II. Ballantine's Illustrated History of World War II *Beekman History of the Wars Library *Barrie Pitt (Consultant ed). The Military History of World War II. Temple Press. London. The Military Press. New York. 1986. [[Category:History]] 0xz928zh2k5our28cebmwo3awootiwn 0 269235 2844157 2762648 2026-11-03T09:55:43Z Jtneill 10242 /* Summary */ Revise introduction 2844157 wikitext text/x-wiki {{title|Nuggets of truth}} This page synthesises the main insights about motivation and emotion report to [[Motivation and emotion|motivation and emotion]] students, as well as providing the original, individual insights. == Summary == This section synthesises the main insights about motivation and emotion reported by previous students (update based on 2024 feedback). These themes capture the most memorable and meaningful principles for understanding and applying the psychology of motivation and emotion identified by students. === Motivation === * '''Basic psychological needs''' - Supporting autonomy, competence, and relatedness fosters intrinsic motivation, engagement, and well-being * '''Intrinsic motivation''' – Nurturing personal interests unlocks intrinsic motivation which sustains goal pursuit * '''Unconscious motivation / Implicit motives''' – Behaviour is often guided by unconscious needs (e.g., for achievement, affiliation, and power) more than logical, conscious intent * '''Growth mindset''' – See ability as developable and embrace challenges and feedback to fuel adaption and growth * '''Goal congruence''' - Pursuing personally meaningful, challenging goals enhances focus and fulfilment === Emotion === * '''Emotion is adaptive''' – Emotions evolved to support survival and social bonding; none are inherently negative * '''Appraisal processes''' – Cognitive interpretation shapes emotional experience; reframing can shift response * '''Emotions are social''' - Emotions create, maintain, and develop social bonds * '''Emotional intelligence''' – Recognising and managing/responding to emotions of self and others * '''Cultivate happiness''' - Happiness emerges from intentional practices aligned with higher personal values/meaning === Applications === * '''Put theory into action''' – Apply motivational and emotional principles to enhance performance and well-being * '''Goal pursuit''' – Specific, challenging, self-congruent goals plus if-then implementation plans faciliate engagement and achievemet * '''Integrated systems''' – Motivation, emotion, and cognition function as interdependent processes * '''Autonomy support''' – Key motivational lever; maximise individual freedom within structural boundaries to optimise motivation and learning; too much or little freedom creates motivational problems === Metacognitive reflections === * '''Contextual nuance''' – Motivation and emotion vary across individuals and contexts; mini-theories offer precision * '''Interdisciplinary scope''' – Core principles of motivation and emotion apply across education, clinical, health, and professional contexts * '''Lifelong learning''' – Integrating knowledge of motivation and emotion insight deepens personal and professional growth == 2024 == === Motivation === * Intrinsic and extrinsic motivation: I find myself applying this to real-world situations all the time now since learning these concepts * Self determinism - competence, relatedness and autonomy * Due to autonomy, it’s hard to get someone to do something if they perceive it as being forced to * Expectancy and control: we make so many choices daily and sometimes don’t realise we go through the process of efficacy expectations and outcome expectations for quick decisions opposed to harder decisions that take longer to decide on. * The role of self efficacy in motivation and intention implementation === Emotion === * Psychological reactance: an unpleasant motivational arousal that emerges when people experience a threat to or loss of their free behaviours. * We feel emotions but they are also a motivator of behaviour - to change / understand situations and our perspectives of things. * Interest and contempt are core emotions? Also surprise sometimes? === Other === * After a while I have an idea about the unit motivation and emotion: maybe one day psychology will have a unit of motivation and a unit of emotion, making each of them significantly standing out as a seperate unit, like psychopathology or social psychology. Maybe the unit of emotion can cover two main chunks, one is about growth psychology (the positive consequences of emotions) and another about psychopathology (the negative consequences of emotions). == 2023 == ===Motivation=== #People are driven by implicit motives for achievement, affiliation, and/or power which can be viewed as acquired psychological needs [[User:U3168525|U3168525]] ([[User talk:U3168525|discuss]] • [[Special:Contributions/U3168525|contribs]]) #Dopamine motivates us in strange ways, be careful how it motivates you. [[User:Yaruki91|U3220114]] #Most optimal freedom for motivation is supporting autonomy and direction within a boundary/constraint. [[User:U3217955|u3217955]] #In regard to what motivates people to believe in conspiracy theories the 2 take-home messages that I found to be most profound were: ##That psychological needs, such as the desire for autonomy, competence, and relatedness, drive individuals to seek explanations that provide a sense of control and understanding, particularly during times of uncertainty and distress. ##That social identity and group dynamics contributes as a motivating factor, as people tend to adopt conspiracy theories to strengthen their sense of belonging or to defend their in-group, especially when they’ve experienced social exclusion. [[User:U3223114|U3223114]] ([[User talk:U3223114|discuss]] • [[Special:Contributions/U3223114|contribs]]) #Continued idea that everyone is different, so exposure to all the varied mini-theories has helped me to feel better equipped to apply and assist, where the general theories had not so far provided enough distinction for me to apply effectively. [[User:JorjaFive|JorjaFive]] ([[User talk:JorjaFive|discuss]] • [[Special:Contributions/JorjaFive|contribs]]) #Learning that motivation can be increased by fulfilling your psychological needs (automony, competence and relatedness) is useful for starting to new projects/jobs etc. [[User:Josh.Cattle|u3226381]]. # Fascinating to trace the roots of components in self-determination theory. Relatedness for example, traces back to the work of Karen Horney’s work on relatedness, and links across to attachment (part of object relations theory), with further links to neuro & oxytocin. It’s a web of roots – and just one part of SDT!! Richard Ryan and Edward Deci are growing what may become an old-growth forest giant [[User:U891419|U891419]]. #The time perspective concepts-> really thought more about how I view my past and if that’s actually how I want to view it (it isn't) so I have started focusing on ways I can alter this moving forward. It also gave me some insight into how I move through time in the present and I've realised I'm prone to living in the past and future so that grounding technique mentioned about doing something for each of your senses everyday has proven to be really helpful! #I found the difference between extrinsic and intrinsic motivation intriguing. Understanding whether motivation comes from an external force, a desire to do the right thing or to please someone else as opposed to doing out of interest or passion gives us a lot of insight into our own interests. ===Emotion=== #I have learned through this unit that there is no such thing as a bad emotion, they all have their own purpose for our survival. #Pride can be separated into authentic pride; pride in one's success, promotes prosocial behaviour, or hubris pride; overconfidence, arrogance, manipulation of others, promotes antisocial behaviour [[User:U3168525|U3168525]] ([[User talk:U3168525|discuss]] • [[Special:Contributions/U3168525|contribs]]) #Synthetic happiness, the idea that people have the ability to create happiness and contentment even in situations that may not naturally lead to happiness, and the concept that our brain systematically misjudges what will make us happy, is fascinating! [[User:U3223114|U3223114]] ([[User talk:U3223114|discuss]] • [[Special:Contributions/U3223114|contribs]]) #Feeling anger is normal. Remember this when having kids so they understand it is ok to be angry, we just don't let our anger hurt others. [[User:Yaruki91|U3220114]] #It pays for communities to possess the resources and social connectedness they need to be resilient. However, we must also address the issues populations face so that the onus is not only on them to cope. [[User:U3213682|U3213682]] ([[User talk:U3213682|discuss]] • [[Special:Contributions/U3213682|contribs]]) 02:42, 25 October 2023 (UTC) #The adaptive purpose of emotions has been really interesting. Learning this makes me feel as though you can use it to your advantage in terms of emotional literacy gains. [[User:Josh.Cattle|u3226381]] #I found the cognitive appraisal approach to emotional experience quite interesting. I have started noticing how people react very differently to the same event. I suppose the positive element about this process is that we have some ability to change the way we feel by challenging the way we see the situation. #That emotions, even one which are universally perceived as negative are evolutionarily designed to aid in survival. ==Process== # Continued idea that everyone is different, so the exposure to all the varied mini-theories has helped me feel better equipped to apply and assist, where the main theories had not so far provided enough distinction # I really enjoyed the book chapter and wikiveristy. I was a little apprehensive at the start, and it seemed overwhelming, but I found it engaging and it was really cool to see other pages and learn a little bit about a lot of things. == 2022 == ===Motivation=== # Motivations and emotions are closely intertwined. # Motivations and emotions link to other psychological disciplines (e.g., health, organisational etc.), not just clinical psychology. ===Emotion=== # Emotions are a big and complex process which deserves as much respect and understanding as thoughts and other mental processes - don't sweep emotions under the proverbial rug; listen to yourself. # <nowiki>Understanding the significance of microexpressions in detecting leakage is essential for emotional intelligence and deception detection. Following the lecture, I began watching the "Lie to Me" series to better understand how the display of personal emotions indicates an individual's intentions in a social situation. </nowiki> [[User:U3211603|U3211603]] ([[User talk:U3211603|discuss]] • [[Special:Contributions/U3211603|contribs]]) 04:11, 15 November 2022 (UTC) # When we don't get what we want, we produce synthetic happiness; this is similar to learned helplessness. It was fascinating to learn about how immersive feelings can make an individual believe that everything is fine on the inside through the use of synthetic happiness. [[User:U3211603|U3211603]] ([[User talk:U3211603|discuss]] • [[Special:Contributions/U3211603|contribs]]) 04:11, 15 November 2022 (UTC) ===Process=== # Treat the chapter and presentation as one assessment piece. By that, complete both, before the former's due date. In honing the latter, I realised the former's focus wasn't sharpened and articulated well enough. # Start early, and continue researching throughout writing your chapter. Don't just stick to the first couple of articles you found in your first search. # Its okay to completely change your chapter. It may feel big and scary at the time but sometimes its necessary. # Monitor your word limit as you progress through writing the book chapter. I found it so easy to just keep writing as I found new and interesting information however, I ended up having to remove a lot of content as a result of being 2000 words of the limit. # I loved the Grades page for Motivation and Emotion unit as it clearly gave me an idea of the progress on the overall grades. It gave me the motivation to work harder and keep my grades good. It would be super nice if this becomes a standard across other unit pages as well. There won't be any need to do any grade calculation. [[User:U3211603|U3211603]] ([[User talk:U3211603|discuss]] • [[Special:Contributions/U3211603|contribs]]) 04:08, 15 November 2022 (UTC) # Discussion in the tutorials has assisted me in applying motivation and emotion theories to real-world problems. Thank you [[User:Jtneill|Jtneill]] for those engaging conversations and for assisting in broadening the horizon. [[User:U3211603|U3211603]] ([[User talk:U3211603|discuss]] • [[Special:Contributions/U3211603|contribs]]) 04:18, 15 November 2022 (UTC) == 2021 == ===Motivation=== # A lot of motivators in society are extrinsic (e.g., legal systems, schooling etc.) # Using extrinsic motivators to try and foster development can be counterproductive. # Intrinsic motivation is REALLY diminished by extrinsic motivation/incentives # Extrinsic vs intrinsic motivation - how they correlate, also trying to use intrinsic motivators for trivial things like trying to go to the gym. # Autonomy, competence, relatedness. Very interesting and ties into so many things. # Connect with others. # Low and high achievers have differences in taking responsibility for successes and failures - an aspect I wasn't aware of. # Growth mindset vs. fixed mindset - become aware of own fixed mindset and been moving towards a more growth-oriented mindset # Gained insight into how people are primarily driven by what matters to them most - maintaining a consistent self-image (e.g., meaning of having to put in effort academically - in a growth mindset it does not mean low intelligence, but learning). # Learned helplessness # Motivation is 'individualistic' in the sense that a lot of it is self-directed and self-motivated. Even if there are social/cultural influences, a lot of how we understand motivation is based on individual perceptions in relation to those social/cultural factors. # Positive psychology # Unconscious motivation #So many interesting facts about motivation (Intrinsic vs extrinsic) to acknowledge, understand and adopting in real world. It is very fascinating. #There are many elements and factors involved in this area, and all are important and relevant for survival and growth #We like to think that we are rational even when we are not. ===Emotion=== # Emotions are a broad topic and there are lots of different types of emotions. # The basic emotions and the specific roles they have; no emotion is unuseful. # Eudaimonic well-being such as life pursuits and quality relationships. # People who experience emotions can be nonverbal and verbal. # Terror management theory: Death and conscious awareness. #Emotions are a great way of expressing feeling from inside your body to outside world. #Humans are not the only ones who experience emotions, it is also present in animals as well! It is such a complex area of study but yet very fascinating. #Optimism by definition describes a certain degree of irrationality in the way we see and think. However, perhaps the only way in which to create a fundamentally better reality is through this irrationality and the corresponding belief that we are good (and getting better)! ===Process=== # Experiment with other platforms to see how it's useful. # Find the leaders in the research - to see what they're saying on the topic - reach out for feedback # Re: book chapter - it's great to start from a point of curiousity in selecting a topic as it will likely be most rewarding. #Learning such an amazing artwork such as wikiversity. very rewarding for present and future as well. #Take home message: Never stop learning and putting your ideas out there because you could be the reason a major breakthrough occurs and makes the world a better place. #Career insight: this unit inspired me to consider a job as a professor/lecturer. I loved how exciting the content was and how this was translated into cool assignments. I also now want to write a whole textbook; but maybe one step at a time… == 2020 == ===Motivation=== # What makes for a good day - to meet needs in Self determination theory # Psychological needs (autonomy/competence/relatedness) and seeing how they look in the workplace # Intrinsic motivation leads to better performance and persistence # Being able to understand what motivates me and relate to content personally # I didn’t realise how often I don’t satisfy my psychological needs in a day (feeling incompetent with uni, not having a choice over what I do because of either work or uni, having to isolate myself to get work done) # You should be trying to incorporate satisfaction of your psychological needs into everyday life, e.g. giving yourself a choice, partaking in something you will be able to feel competent in, and connecting with others. # How motivation influences everything we do, and how we can optimize it # Explanatory styles: much of motivation is not so much about reality, than perceptions. Negative perceptions undermine performance and happiness. # Being controlling when someone is being disruptive is detrimental to their motivation and can actually make them more defiant!! # Expression of motivation # Optimism can be learned, and having a belief in perceived control can help with a growth mindset # Cognitive evaluation theory - explains how all events have two functions (controlling behaviour and informing competence) # The relationship between attitudes and motivation # Maslow’s hierarchy of needs - suggesting that humans can be organised in a hierarchy (physiological and then psychological) # The large part that motivation plays into our cognitions and it’s level of importance. # Formalising intrinsic and extrinsic thought (SDT) # Extrinsic and intrinsic motivators. How to use extrinsic effectively and the importance of intrinsic in motivation. # We have autonomy, competence, relatedness needs # Partnership between goals, needs and well-being. # Learning about mastery vs. performance goals made me reassess my approach to uni -- previously used to approach it as “I can get better grades than my siblings to show everyone I’m the better child”, but have now approached it in a way that is more focused on what I’m actually trying to achieve. # That Ted talk about the people with amnesia who still preferred their painting was wild # Power needs, affiliation/relatedness needs, achievement needs # Choice is vital to motivation; students who are allowed agency to choose assignment topics tend to feel more motivated. # Working towards having mastery goals (rather than performance goals) and a growth mindset (rather than a fixed one) is an incredibly worthwhile pursuit - both for one’s well-being and success. # It’s important that your goals are both challenging and within your abilities. # A lot of motivation is unconscious. # Performance can be enhanced by being selective about goals rather than having too many goals. # Goals aligned with personal values are congruent and easier to pursue than incongruent goals which are not aligned with personal values. # Learning how to activate flow states and experience them more often will benefit both your mental health and the quality of your work. ===Emotion=== # "Happiness is not something ready made. It comes from your own actions." Dalai Lama XIV. Linking with Reeve chapter 17: motivation and emotion are flexible to apply, but require long-term application. # Take-home from [[Motivation and emotion/Book/2020/Workplace mental health|my book chapter]]: being congruent. Considering our life as a whole: our working/studying life and home-life being complementary or leading to conflict. Reflecting on Ryff's 1995, 6 facets of well-being: self-acceptance, interpersonal relations, autonomy, environmental mastery, purpose in life and personal growth and how they all link together. The outcomes of being incongruent may lead to psychological distress and how often do we not realise this link? # All emotions are good and have a meaning/reason for why we feel them # Incorporating compassion into your everyday life can impact you in the best possible ways. Feelings of guilt, shame and other negative emotions diminish. # Increasing your emotional vocabulary helps you understand how you’re feeling. # All our emotions have evolved to help us # That we should experience emotions, but work towards managing them effectively. # The four components of emotion and how they work together to influence our emotions. And that we can make changes by intercepting at these points. # Core emotions, self conscious emotions , and the cognitively complex ones # The most significant thing for me when it came to emotion was that for once the psychodynamic perspective actually sounded reasonable. # We underestimate well-being and the effects on/in our life. # All emotions serve a purpose - negative emotions are just as valuable as positive ones. # Part of emotional literacy is learning to become friends with all of our emotions - they have a place in our lives. # One of the strategies for enhancing happiness is to learn how to broaden-and-build on the experience of positive emotions. # A lot of our emotional lives are triggered by encounters with other people. # One of the most challenging emotions to learn is compersion (rather than jealousy) - feeling good about someone else’s pleasure. ===Other=== # Delving into motivation and emotion has given me more to reflect on. Tutorials and lectures were very interesting and useful in general. [[wikipedia:Self-determination_theory|SDT]] - intrinsic and extrinsic motivators, James said this theory was the most applicable and I now see why! # TED talks and similar have been great. They have provided further insights into a very interesting and thought-provoking unit. I have forwarded them on to others (not involved in psychology). Thank you James. ==2019== # Approach behaviour is stronger than avoidance behaviour. # "Between stimulus and response there is a space. In that space is our power to choose that response. In our response lies our growth and our freedom." (Viktor E. Frankl) # Exercise regularly - it has many health benefits including enhancing mood. # Focus on intrinsic motivation more and extrinsic less. # In the face of challenges, keep going. # Intrinsic motivation stronger than extrinsic motivation. # In workplaces and education: encourage autonomy, competence, relatedness. # Motivation can be both conscious and unconscious. We aren't always aware of our motivations. # People who feel they have no control, or little control, over their environment don't fare well! # Positive moods enhance creativity on short tasks, negative emotions enhance creativity on tasks of longer duration. # Thrive on your success. # When you achieve something really acknowledge it! Self-efficacy. ==2018== # By identifying how we respond to unexpected changes we may influence our emotional reactions and better adapt to our environment. #Designing and writing a book chapter on a topic you're interested in was a valuable experience that I enjoyed and I'm proud of. # Emotions vs. mood. # Even when people are in bad/unpleasant situations, being able to change the situation so they can become more happy isn't as simple as it seems. # Goal-setting and goal-striving: gave me some tips for my own life # [https://www.ted.com/talks/kelly_mcgonigal_how_to_make_stress_your_friend? How to make stress your friend language=en] - Really enjoyed this ted talk from earlier on in the semester # I want a carrot for this comment James # Ideologies - how these can motivate people through humanistic psychology and how people are "evil". # Impacts of goal sharing (findings from book chapter). # Intrinsic motivation is a more effective form of studying. Focusing on things you're interested in increases productivity. # Mindfulness has a significant effect in psychological therapy (from book chapter). # Motivation is private and can usually only come from yourself. # Self-awareness about link between job choice (interests) and personality and need satisfaction (book chapter). # The surprising science of happiness (TED talk in Tutorial 6). # The work done in the book chapter - will take its knowledge into my profession. # Time perspective has a significant influence on the human experience. ==2016-2017== # [https://padlet.com/jamestneill/i7485qnwioeu Pearls of wisdom / Nuggets of truth 2016-2017: What we learnt from this unit] (padlet.com) ==2015== # [http://padlet.com/jamestneill/ahf1922g2958 Pearls of wisdom / Nuggets of truth 2015: What we learnt from this unit] (padlet.com) ==2014== # [http://padlet.com/jamestneill/dirbzepdgpsl Pearls of wisdom / Nuggets of truth 2014: What we learnt from this unit] (padlet.com) ==2013== # [http://padlet.com/jamestneill/s7g2hu1zc Pearls of wisdom / Nuggets of truth 2013: What we learnt from this unit] (padlet.com) ==2011== # [[Motivation and emotion/Book/2011/Pearls of wisdom|2011]] [[Category:Motivation and emotion]] czs8yg6o3qvabk3k5xqse7uykshzjd6 2844158 2844157 2026-11-03T09:58:41Z Jtneill 10242 + 2025 list of insights 2844158 wikitext text/x-wiki {{title|Nuggets of truth}} This page synthesises the main insights about motivation and emotion report to [[Motivation and emotion|motivation and emotion]] students, as well as providing the original, individual insights. The first part summarises the student insights into themes. The second part provides the individual insights by year. == Summary == This section synthesises the main insights about motivation and emotion reported by previous students (update based on 2024 feedback). These themes capture the most memorable and meaningful principles for understanding and applying the psychology of motivation and emotion identified by students. === Motivation === * '''Basic psychological needs''' - Supporting autonomy, competence, and relatedness fosters intrinsic motivation, engagement, and well-being * '''Intrinsic motivation''' – Nurturing personal interests unlocks intrinsic motivation which sustains goal pursuit * '''Unconscious motivation / Implicit motives''' – Behaviour is often guided by unconscious needs (e.g., for achievement, affiliation, and power) more than logical, conscious intent * '''Growth mindset''' – See ability as developable and embrace challenges and feedback to fuel adaption and growth * '''Goal congruence''' - Pursuing personally meaningful, challenging goals enhances focus and fulfilment === Emotion === * '''Emotion is adaptive''' – Emotions evolved to support survival and social bonding; none are inherently negative * '''Appraisal processes''' – Cognitive interpretation shapes emotional experience; reframing can shift response * '''Emotions are social''' - Emotions create, maintain, and develop social bonds * '''Emotional intelligence''' – Recognising and managing/responding to emotions of self and others * '''Cultivate happiness''' - Happiness emerges from intentional practices aligned with higher personal values/meaning === Applications === * '''Put theory into action''' – Apply motivational and emotional principles to enhance performance and well-being * '''Goal pursuit''' – Specific, challenging, self-congruent goals plus if-then implementation plans faciliate engagement and achievemet * '''Integrated systems''' – Motivation, emotion, and cognition function as interdependent processes * '''Autonomy support''' – Key motivational lever; maximise individual freedom within structural boundaries to optimise motivation and learning; too much or little freedom creates motivational problems === Metacognitive reflections === * '''Contextual nuance''' – Motivation and emotion vary across individuals and contexts; mini-theories offer precision * '''Interdisciplinary scope''' – Core principles of motivation and emotion apply across education, clinical, health, and professional contexts * '''Lifelong learning''' – Integrating knowledge of motivation and emotion insight deepens personal and professional growth ==2025== ===Motivation=== * The use of autonomy in helping or guiding others. I have noticed that when I give my kids more choices it has resulted in better results for all of us * Something that unexpectedly resonated with me was fixed-growth mindsets! Having found I lean towards the former, finding out that it and its associated performance goals tend to be unhealthy and unsustainable. Trying to move toward more of a growth-mindset now! * Part of my book chapter found that a lot of traditionally used motivators can actually be really detrimental to motivation, or at least, optimal motivation, because they diminish autonomy. So having autonomy is really key for the best kind of motivation. * That people need autonomy so much that we will engage in psychological reactance to reassert that autonomy even when it may not be in our best, long-term interests * I’ve also tried to be more intrinsically motivated about tasks, and “find the fun”, so to speak. * Framing all behaviours as some degree of motivation was really interesting and thinking about “why am I doing this behaviour?”. Not just the super energetic ones that are typically framed as being motivated, but also things like “why am I motivated to just stay in bed this morning?”. * The neuroanatomy and neurotransmitters involved in motivation. * Understanding how being neurodivergent can impact motivation. * Exploring psychological needs * Learning about mindsets - cognitive patterns or mental frameworks and how mindsets influence thinking, behaviours and feelings * This unit solidified for me that motivation and the factors/ triggers that cause someone to feel motivated or demotivated are so unique to a person such as in the beginning of the unit we discussed two different reasons for people to go for a run in the morning for person 1 is training for a marathon or running to keep fit while person 2 is going for a run to catch up with friends or as a social outing. * Learning motivation helps us to understand what we want, why we want and how. * Understanding how relatedness, autonomy and competence can be applied in an organisational or work setting to achieve a certain outcome. * Learning about implicit motives. It made me realise that motives aren’t always something we are consciously aware of, and it was quite fascinating noticing that I am predominantly driven by affiliation. * This class reaffirmed a lot of the things I knew about mindsets and how they can shape our mental health * Motivation is both an internal individual process as well as an external influenced process * Freud’s dual instinct theory - Eros (life instinct) and Thanatos (death drive) - positive vs destructive coping mechanisms to unconscious instincts * I really enjoyed this unit as a whole especially thought it was interesting to learn about the nucleus accumbens role in motivation. * For me it was learning about self determination theory and applying it into the way I motivate myself and others. In particular the ideas around relatedness, autonomy and competence. ===Emotion=== * I was surprised that interest was an emotion. I had never thought of interest in this way but now I find I utilise it in daily activities. For example in revision I try and find little topics of interest that encourage more reading/learning * Learning about the vast amount of emotions and understanding the differences between some and how to distinguish between them * Learning about emotional intelligence and it being divided by awareness and management and self vs. social or relational. * It was interesting learning different definitions for both basic and complex emotions as typically I wouldn’t describe them as the provided definition eg. anger, however when applied to the context of motivations of behaviour, it all makes sense and clicks better for me * Understanding how being neurodivergent can impact emotions. * Emotions are not as easy to categorise as I thought * I learnt not to internalise emotions or label them too quickly * All emotions have value and purpose and can be leveraged to help with motivational endeavours. Anxiety can be reappraised as excitement. * Learning from my book chapter helped me bridge emotion and motivation together and look at their relationship differently * It was interesting learning about the distinction between basic, cognitively complex, and self-conscious emotions. * How each emotion has a communicative function, which is recognisable across cultures. * Human emotion is a very tricky thing. Emotion is the most interesting part of Psychology I would like to dig deeper into. * Emotions can be motivators * Schadenfruede - evil joy :) * The power of synthetic happiness is astonishing! ===Other=== * Really liked the time perspective tutorial and this is something I’ve been considering in my day to day life since * I’m interested in learning more about positive psychology. I think it’s so important to not merely look at people from a deficit point of view. Don’t just look at what’s wrong, look at what’s going right! * I have found positive psychology really interesting. I * Learning how to use wikiversity and learning so much from the bookchapters * I would be interested to learn more about the time perspective as it was something I didn’t know much about and have tried applying it to everyday life and shifting my perspectives of things to have an equal balance as to which point in time i live in more than another. * Learning about the emphasis of meaning and purpose in positive psychology was a really nice reminder of finding a way to make things feel meaningful where I can. * I found it really interesting taking the activities and questionnaires from tutorials and applying it to my friends, specifically the time constraints we looked at. Getting them to complete them and predict their scores based on what I know of their personality was really fun and helped to apply it and see it among people I know. * Learning the importance of hobbies in facilitating well-being really stuck with me. * The influence of personality on these aspects was something I wasn’t aware of prior * Understanding how the time perspective can influence positive and negative outcomes in a person's life, as well as how positive psychology provides a more holistic view of psychology, which is often not discussed in much depth. * Self-actualising isn’t just becoming really good at something. It can expand and reflect outwards for the greater good, and help others reach their optimal being. == 2024 == === Motivation === * Intrinsic and extrinsic motivation: I find myself applying this to real-world situations all the time now since learning these concepts * Self determinism - competence, relatedness and autonomy * Due to autonomy, it’s hard to get someone to do something if they perceive it as being forced to * Expectancy and control: we make so many choices daily and sometimes don’t realise we go through the process of efficacy expectations and outcome expectations for quick decisions opposed to harder decisions that take longer to decide on. * The role of self efficacy in motivation and intention implementation === Emotion === * Psychological reactance: an unpleasant motivational arousal that emerges when people experience a threat to or loss of their free behaviours. * We feel emotions but they are also a motivator of behaviour - to change / understand situations and our perspectives of things. * Interest and contempt are core emotions? Also surprise sometimes? === Other === * After a while I have an idea about the unit motivation and emotion: maybe one day psychology will have a unit of motivation and a unit of emotion, making each of them significantly standing out as a seperate unit, like psychopathology or social psychology. Maybe the unit of emotion can cover two main chunks, one is about growth psychology (the positive consequences of emotions) and another about psychopathology (the negative consequences of emotions). == 2023 == ===Motivation=== #People are driven by implicit motives for achievement, affiliation, and/or power which can be viewed as acquired psychological needs [[User:U3168525|U3168525]] ([[User talk:U3168525|discuss]] • [[Special:Contributions/U3168525|contribs]]) #Dopamine motivates us in strange ways, be careful how it motivates you. [[User:Yaruki91|U3220114]] #Most optimal freedom for motivation is supporting autonomy and direction within a boundary/constraint. [[User:U3217955|u3217955]] #In regard to what motivates people to believe in conspiracy theories the 2 take-home messages that I found to be most profound were: ##That psychological needs, such as the desire for autonomy, competence, and relatedness, drive individuals to seek explanations that provide a sense of control and understanding, particularly during times of uncertainty and distress. ##That social identity and group dynamics contributes as a motivating factor, as people tend to adopt conspiracy theories to strengthen their sense of belonging or to defend their in-group, especially when they’ve experienced social exclusion. [[User:U3223114|U3223114]] ([[User talk:U3223114|discuss]] • [[Special:Contributions/U3223114|contribs]]) #Continued idea that everyone is different, so exposure to all the varied mini-theories has helped me to feel better equipped to apply and assist, where the general theories had not so far provided enough distinction for me to apply effectively. [[User:JorjaFive|JorjaFive]] ([[User talk:JorjaFive|discuss]] • [[Special:Contributions/JorjaFive|contribs]]) #Learning that motivation can be increased by fulfilling your psychological needs (automony, competence and relatedness) is useful for starting to new projects/jobs etc. [[User:Josh.Cattle|u3226381]]. # Fascinating to trace the roots of components in self-determination theory. Relatedness for example, traces back to the work of Karen Horney’s work on relatedness, and links across to attachment (part of object relations theory), with further links to neuro & oxytocin. It’s a web of roots – and just one part of SDT!! Richard Ryan and Edward Deci are growing what may become an old-growth forest giant [[User:U891419|U891419]]. #The time perspective concepts-> really thought more about how I view my past and if that’s actually how I want to view it (it isn't) so I have started focusing on ways I can alter this moving forward. It also gave me some insight into how I move through time in the present and I've realised I'm prone to living in the past and future so that grounding technique mentioned about doing something for each of your senses everyday has proven to be really helpful! #I found the difference between extrinsic and intrinsic motivation intriguing. Understanding whether motivation comes from an external force, a desire to do the right thing or to please someone else as opposed to doing out of interest or passion gives us a lot of insight into our own interests. ===Emotion=== #I have learned through this unit that there is no such thing as a bad emotion, they all have their own purpose for our survival. #Pride can be separated into authentic pride; pride in one's success, promotes prosocial behaviour, or hubris pride; overconfidence, arrogance, manipulation of others, promotes antisocial behaviour [[User:U3168525|U3168525]] ([[User talk:U3168525|discuss]] • [[Special:Contributions/U3168525|contribs]]) #Synthetic happiness, the idea that people have the ability to create happiness and contentment even in situations that may not naturally lead to happiness, and the concept that our brain systematically misjudges what will make us happy, is fascinating! [[User:U3223114|U3223114]] ([[User talk:U3223114|discuss]] • [[Special:Contributions/U3223114|contribs]]) #Feeling anger is normal. Remember this when having kids so they understand it is ok to be angry, we just don't let our anger hurt others. [[User:Yaruki91|U3220114]] #It pays for communities to possess the resources and social connectedness they need to be resilient. However, we must also address the issues populations face so that the onus is not only on them to cope. [[User:U3213682|U3213682]] ([[User talk:U3213682|discuss]] • [[Special:Contributions/U3213682|contribs]]) 02:42, 25 October 2023 (UTC) #The adaptive purpose of emotions has been really interesting. Learning this makes me feel as though you can use it to your advantage in terms of emotional literacy gains. [[User:Josh.Cattle|u3226381]] #I found the cognitive appraisal approach to emotional experience quite interesting. I have started noticing how people react very differently to the same event. I suppose the positive element about this process is that we have some ability to change the way we feel by challenging the way we see the situation. #That emotions, even one which are universally perceived as negative are evolutionarily designed to aid in survival. ==Process== # Continued idea that everyone is different, so the exposure to all the varied mini-theories has helped me feel better equipped to apply and assist, where the main theories had not so far provided enough distinction # I really enjoyed the book chapter and wikiveristy. I was a little apprehensive at the start, and it seemed overwhelming, but I found it engaging and it was really cool to see other pages and learn a little bit about a lot of things. == 2022 == ===Motivation=== # Motivations and emotions are closely intertwined. # Motivations and emotions link to other psychological disciplines (e.g., health, organisational etc.), not just clinical psychology. ===Emotion=== # Emotions are a big and complex process which deserves as much respect and understanding as thoughts and other mental processes - don't sweep emotions under the proverbial rug; listen to yourself. # <nowiki>Understanding the significance of microexpressions in detecting leakage is essential for emotional intelligence and deception detection. Following the lecture, I began watching the "Lie to Me" series to better understand how the display of personal emotions indicates an individual's intentions in a social situation. </nowiki> [[User:U3211603|U3211603]] ([[User talk:U3211603|discuss]] • [[Special:Contributions/U3211603|contribs]]) 04:11, 15 November 2022 (UTC) # When we don't get what we want, we produce synthetic happiness; this is similar to learned helplessness. It was fascinating to learn about how immersive feelings can make an individual believe that everything is fine on the inside through the use of synthetic happiness. [[User:U3211603|U3211603]] ([[User talk:U3211603|discuss]] • [[Special:Contributions/U3211603|contribs]]) 04:11, 15 November 2022 (UTC) ===Process=== # Treat the chapter and presentation as one assessment piece. By that, complete both, before the former's due date. In honing the latter, I realised the former's focus wasn't sharpened and articulated well enough. # Start early, and continue researching throughout writing your chapter. Don't just stick to the first couple of articles you found in your first search. # Its okay to completely change your chapter. It may feel big and scary at the time but sometimes its necessary. # Monitor your word limit as you progress through writing the book chapter. I found it so easy to just keep writing as I found new and interesting information however, I ended up having to remove a lot of content as a result of being 2000 words of the limit. # I loved the Grades page for Motivation and Emotion unit as it clearly gave me an idea of the progress on the overall grades. It gave me the motivation to work harder and keep my grades good. It would be super nice if this becomes a standard across other unit pages as well. There won't be any need to do any grade calculation. [[User:U3211603|U3211603]] ([[User talk:U3211603|discuss]] • [[Special:Contributions/U3211603|contribs]]) 04:08, 15 November 2022 (UTC) # Discussion in the tutorials has assisted me in applying motivation and emotion theories to real-world problems. Thank you [[User:Jtneill|Jtneill]] for those engaging conversations and for assisting in broadening the horizon. [[User:U3211603|U3211603]] ([[User talk:U3211603|discuss]] • [[Special:Contributions/U3211603|contribs]]) 04:18, 15 November 2022 (UTC) == 2021 == ===Motivation=== # A lot of motivators in society are extrinsic (e.g., legal systems, schooling etc.) # Using extrinsic motivators to try and foster development can be counterproductive. # Intrinsic motivation is REALLY diminished by extrinsic motivation/incentives # Extrinsic vs intrinsic motivation - how they correlate, also trying to use intrinsic motivators for trivial things like trying to go to the gym. # Autonomy, competence, relatedness. Very interesting and ties into so many things. # Connect with others. # Low and high achievers have differences in taking responsibility for successes and failures - an aspect I wasn't aware of. # Growth mindset vs. fixed mindset - become aware of own fixed mindset and been moving towards a more growth-oriented mindset # Gained insight into how people are primarily driven by what matters to them most - maintaining a consistent self-image (e.g., meaning of having to put in effort academically - in a growth mindset it does not mean low intelligence, but learning). # Learned helplessness # Motivation is 'individualistic' in the sense that a lot of it is self-directed and self-motivated. Even if there are social/cultural influences, a lot of how we understand motivation is based on individual perceptions in relation to those social/cultural factors. # Positive psychology # Unconscious motivation #So many interesting facts about motivation (Intrinsic vs extrinsic) to acknowledge, understand and adopting in real world. It is very fascinating. #There are many elements and factors involved in this area, and all are important and relevant for survival and growth #We like to think that we are rational even when we are not. ===Emotion=== # Emotions are a broad topic and there are lots of different types of emotions. # The basic emotions and the specific roles they have; no emotion is unuseful. # Eudaimonic well-being such as life pursuits and quality relationships. # People who experience emotions can be nonverbal and verbal. # Terror management theory: Death and conscious awareness. #Emotions are a great way of expressing feeling from inside your body to outside world. #Humans are not the only ones who experience emotions, it is also present in animals as well! It is such a complex area of study but yet very fascinating. #Optimism by definition describes a certain degree of irrationality in the way we see and think. However, perhaps the only way in which to create a fundamentally better reality is through this irrationality and the corresponding belief that we are good (and getting better)! ===Process=== # Experiment with other platforms to see how it's useful. # Find the leaders in the research - to see what they're saying on the topic - reach out for feedback # Re: book chapter - it's great to start from a point of curiousity in selecting a topic as it will likely be most rewarding. #Learning such an amazing artwork such as wikiversity. very rewarding for present and future as well. #Take home message: Never stop learning and putting your ideas out there because you could be the reason a major breakthrough occurs and makes the world a better place. #Career insight: this unit inspired me to consider a job as a professor/lecturer. I loved how exciting the content was and how this was translated into cool assignments. I also now want to write a whole textbook; but maybe one step at a time… == 2020 == ===Motivation=== # What makes for a good day - to meet needs in Self determination theory # Psychological needs (autonomy/competence/relatedness) and seeing how they look in the workplace # Intrinsic motivation leads to better performance and persistence # Being able to understand what motivates me and relate to content personally # I didn’t realise how often I don’t satisfy my psychological needs in a day (feeling incompetent with uni, not having a choice over what I do because of either work or uni, having to isolate myself to get work done) # You should be trying to incorporate satisfaction of your psychological needs into everyday life, e.g. giving yourself a choice, partaking in something you will be able to feel competent in, and connecting with others. # How motivation influences everything we do, and how we can optimize it # Explanatory styles: much of motivation is not so much about reality, than perceptions. Negative perceptions undermine performance and happiness. # Being controlling when someone is being disruptive is detrimental to their motivation and can actually make them more defiant!! # Expression of motivation # Optimism can be learned, and having a belief in perceived control can help with a growth mindset # Cognitive evaluation theory - explains how all events have two functions (controlling behaviour and informing competence) # The relationship between attitudes and motivation # Maslow’s hierarchy of needs - suggesting that humans can be organised in a hierarchy (physiological and then psychological) # The large part that motivation plays into our cognitions and it’s level of importance. # Formalising intrinsic and extrinsic thought (SDT) # Extrinsic and intrinsic motivators. How to use extrinsic effectively and the importance of intrinsic in motivation. # We have autonomy, competence, relatedness needs # Partnership between goals, needs and well-being. # Learning about mastery vs. performance goals made me reassess my approach to uni -- previously used to approach it as “I can get better grades than my siblings to show everyone I’m the better child”, but have now approached it in a way that is more focused on what I’m actually trying to achieve. # That Ted talk about the people with amnesia who still preferred their painting was wild # Power needs, affiliation/relatedness needs, achievement needs # Choice is vital to motivation; students who are allowed agency to choose assignment topics tend to feel more motivated. # Working towards having mastery goals (rather than performance goals) and a growth mindset (rather than a fixed one) is an incredibly worthwhile pursuit - both for one’s well-being and success. # It’s important that your goals are both challenging and within your abilities. # A lot of motivation is unconscious. # Performance can be enhanced by being selective about goals rather than having too many goals. # Goals aligned with personal values are congruent and easier to pursue than incongruent goals which are not aligned with personal values. # Learning how to activate flow states and experience them more often will benefit both your mental health and the quality of your work. ===Emotion=== # "Happiness is not something ready made. It comes from your own actions." Dalai Lama XIV. Linking with Reeve chapter 17: motivation and emotion are flexible to apply, but require long-term application. # Take-home from [[Motivation and emotion/Book/2020/Workplace mental health|my book chapter]]: being congruent. Considering our life as a whole: our working/studying life and home-life being complementary or leading to conflict. Reflecting on Ryff's 1995, 6 facets of well-being: self-acceptance, interpersonal relations, autonomy, environmental mastery, purpose in life and personal growth and how they all link together. The outcomes of being incongruent may lead to psychological distress and how often do we not realise this link? # All emotions are good and have a meaning/reason for why we feel them # Incorporating compassion into your everyday life can impact you in the best possible ways. Feelings of guilt, shame and other negative emotions diminish. # Increasing your emotional vocabulary helps you understand how you’re feeling. # All our emotions have evolved to help us # That we should experience emotions, but work towards managing them effectively. # The four components of emotion and how they work together to influence our emotions. And that we can make changes by intercepting at these points. # Core emotions, self conscious emotions , and the cognitively complex ones # The most significant thing for me when it came to emotion was that for once the psychodynamic perspective actually sounded reasonable. # We underestimate well-being and the effects on/in our life. # All emotions serve a purpose - negative emotions are just as valuable as positive ones. # Part of emotional literacy is learning to become friends with all of our emotions - they have a place in our lives. # One of the strategies for enhancing happiness is to learn how to broaden-and-build on the experience of positive emotions. # A lot of our emotional lives are triggered by encounters with other people. # One of the most challenging emotions to learn is compersion (rather than jealousy) - feeling good about someone else’s pleasure. ===Other=== # Delving into motivation and emotion has given me more to reflect on. Tutorials and lectures were very interesting and useful in general. [[wikipedia:Self-determination_theory|SDT]] - intrinsic and extrinsic motivators, James said this theory was the most applicable and I now see why! # TED talks and similar have been great. They have provided further insights into a very interesting and thought-provoking unit. I have forwarded them on to others (not involved in psychology). Thank you James. ==2019== # Approach behaviour is stronger than avoidance behaviour. # "Between stimulus and response there is a space. In that space is our power to choose that response. In our response lies our growth and our freedom." (Viktor E. Frankl) # Exercise regularly - it has many health benefits including enhancing mood. # Focus on intrinsic motivation more and extrinsic less. # In the face of challenges, keep going. # Intrinsic motivation stronger than extrinsic motivation. # In workplaces and education: encourage autonomy, competence, relatedness. # Motivation can be both conscious and unconscious. We aren't always aware of our motivations. # People who feel they have no control, or little control, over their environment don't fare well! # Positive moods enhance creativity on short tasks, negative emotions enhance creativity on tasks of longer duration. # Thrive on your success. # When you achieve something really acknowledge it! Self-efficacy. ==2018== # By identifying how we respond to unexpected changes we may influence our emotional reactions and better adapt to our environment. #Designing and writing a book chapter on a topic you're interested in was a valuable experience that I enjoyed and I'm proud of. # Emotions vs. mood. # Even when people are in bad/unpleasant situations, being able to change the situation so they can become more happy isn't as simple as it seems. # Goal-setting and goal-striving: gave me some tips for my own life # [https://www.ted.com/talks/kelly_mcgonigal_how_to_make_stress_your_friend? How to make stress your friend language=en] - Really enjoyed this ted talk from earlier on in the semester # I want a carrot for this comment James # Ideologies - how these can motivate people through humanistic psychology and how people are "evil". # Impacts of goal sharing (findings from book chapter). # Intrinsic motivation is a more effective form of studying. Focusing on things you're interested in increases productivity. # Mindfulness has a significant effect in psychological therapy (from book chapter). # Motivation is private and can usually only come from yourself. # Self-awareness about link between job choice (interests) and personality and need satisfaction (book chapter). # The surprising science of happiness (TED talk in Tutorial 6). # The work done in the book chapter - will take its knowledge into my profession. # Time perspective has a significant influence on the human experience. ==2016-2017== # [https://padlet.com/jamestneill/i7485qnwioeu Pearls of wisdom / Nuggets of truth 2016-2017: What we learnt from this unit] (padlet.com) ==2015== # [http://padlet.com/jamestneill/ahf1922g2958 Pearls of wisdom / Nuggets of truth 2015: What we learnt from this unit] (padlet.com) ==2014== # [http://padlet.com/jamestneill/dirbzepdgpsl Pearls of wisdom / Nuggets of truth 2014: What we learnt from this unit] (padlet.com) ==2013== # [http://padlet.com/jamestneill/s7g2hu1zc Pearls of wisdom / Nuggets of truth 2013: What we learnt from this unit] (padlet.com) ==2011== # [[Motivation and emotion/Book/2011/Pearls of wisdom|2011]] [[Category:Motivation and emotion]] 6xb5ldt5lob7minlcx99879us0bojgy 0 282475 2844177 2817499 2026-11-03T11:23:26Z ThaniosAkro 2805358 /* With no common point */ 2844177 wikitext text/x-wiki The quartic function is the bridge between the cubic function and more advanced functions such as the quintic and sextic. =Objective= {{RoundBoxTop|theme=3}} [[file:Books-aj.svg_aj_ashton_01f.png|right|100px]] * Present quartic function and quartic equation. * Introduce the concept of roots of equal absolute value. * Show how to predict and calculate equal roots, techniques that will be useful when applied to higher order functions. * Simplify the depressed quartic. * Show that the quartic equation is effectively solved when at least one root is known. * Present the "resolvent" cubic function. * Show how to derive and use the quartic formula. {{RoundBoxBottom}} =Lesson= {{RoundBoxTop|theme=2}} ==Introduction== The quartic function is the sum of powers of <math>x</math> from <math>0</math> through <math>4</math>: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx^1 + ex^0</math> usually written as: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx + e.</math> If <math>e == 0</math> the function becomes <math>x(ax^3 + bx^2 + cx + d).</math> Within this page we'll say that: * both coefficients <math>a, e</math> must be non-zero, * coefficient <math>a</math> must be positive (simply for our convenience), * all coefficients must be real numbers, accepting that the function may contain complex roots. The quartic equation is the quartic function equated to zero: <math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>. Roots of the function are values of <math>x</math> that satisfy the quartic equation. {{RoundBoxTop|theme=1}} Because the function is "quartic" (maximum power of <math>x</math> is <math>4</math>), the function contains exactly <math>4</math> roots, an even number of complex roots and an even number of real roots. Other combinations of real and complex roots are possible, but they produce complex coefficients. {{RoundBoxBottom}} {{RoundBoxTop|theme=7}} [[File:0312quartic01a.png|thumb|400px|'''Graph of typical quartic function showing minima and maximum.''']] The figure shows a typical quartic function. The function crosses the <math>X</math> axis in 4 different places. The function has 4 roots: <math>(-2,0), (1,0), (5,0), (10,0).</math> This function contains one local minimum, one local maximum and one absolute minimum. There is no absolute maximum. Because the function contains one absolute minimum: * If abs(<math>x</math>) is very large, <math>f(x)</math> is always positive. * If absolute minimum is above <math>X</math> axis, curve does not cross <math>X</math> axis and function contains only complex roots. * There is always at least one point where the curve is parallel to <math>X</math> axis. The curve is never parallel to the <math>Y</math> axis. For any real value of <math>x</math> there is always a real value of <math>y.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0312quartic02.png|thumb|400px|'''When coefficient d is missing, there is a stationary point at x = 0.''']] If coefficient <math>d</math> is missing, the quartic function becomes <math>y = ax^4 + bx^3 + cx^2 + e,</math> and <math>y' = 4ax^3 + 3bx^2 + 2cx = x(4ax^2 + 3bx + 2c).</math> For a stationary point <math>y' = x(4ax^2 + 3bx + 2c) = 0.</math> When coefficient <math>d</math> is missing, there is always a stationary point at <math>x = 0.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} [[File:0319quartic02.png|thumb|400px|'''Graph of quartic function that is quadratic in <math>x^2</math>.'''</br>Because coefficient <math>d</math> is missing, there is a stationary point where <math>x = 0.</math>]] If coefficients <math>b, d</math> are missing, the quartic function becomes a quadratic in <math>x^2.</math> The curve (red line) in diagram has equation: <math>y = f(x) = \frac{x^4 - 13x^2 + 36}{5}</math> The quartic equation may be solved as: <math>X^2 - 13X + 36 = 0</math> where <math>X = x^2</math> or <math>x = \sqrt{X}.</math> <math>X = 4</math> or <math>X = 9.</math> <math>x = \pm 2</math> or <math>x = \pm 3.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0312quartic03.png|thumb|400px| ]] The quartic function may be expressed as <math>x = ay^4 + by^3 + cy^2 + dy + e.</math> Unless otherwise noted, references to "quartic function" on this page refer to function of form <math>y = ax^4 + bx^3 + cx^2 + dx + e.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0318quartic01.png|thumb|400px|'''Graph of quartic function with coefficient a negative.'''</br>There is no absolute minimum.]] Coefficient <math>a</math> may be negative as shown in diagram. As <code>abs</code><math>(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^4.</math> When <code>abs</code><math>(x)</math> is very large, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "quartic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0502quartic01.png|thumb|400px|'''Graph of quartic function with coefficient b missing.''' </br> Sum of roots is <math>0.</math> </br> <math>Y</math>axis compressed for clarity. ]] When sum of roots is <math>0,</math> coefficient <math>b = 0.</math> In the diagram, roots of <math>f(x)</math> are <math>-5, -4, 2, 7.</math> Sum of roots <math>= 0.</math> Therefore coefficient <math>b = 0.</math> {{RoundBoxBottom}} ==Function as product of linear function and cubic== {{RoundBoxTop|theme=3}} [[File:0313quartic01.png|thumb|400px|'''Graphs of quartic function and associated cubic function.''']] When <math>p</math> is a root of the function, the function may be expressed as: <math>(x - p)( Ax^3 + Bx^2 + Cx + D )</math> where <math>A = a;\ B = Ap + b;\ C = Bp + c;\ D = Cp + d.</math> When one real root <math>p</math> is known, the other three roots may be calculated as roots of the cubic function <math>Ax^3 + Bx^2 + Cx + D.</math> In the diagram the quartic function has equation: <math>y = \frac{ x^4 - 23x^3 + 163x^2 - 393x + 252 }{ 48 }.</math> It is known that <math>3</math> is a root of this function. The associated cubic has equation: <math>y = \frac { x^3 - 20x^2 + 103x - 84} { 48 }</math> The 2 curves coincide at points <math>(1, 0),\ (7, 0),\ (12, 0),</math> the three points that are roots of both functions. <math></math> {{RoundBoxBottom}} ==Function defined by 5 points== {{RoundBoxTop|theme=2}} [[File:0313quartic02.png|thumb|400px|'''Figure 1. Quartic function defined by 5 points''' </br>Any 5 points on the curve may be used to define the function.]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 5 unique points on the curve. For example, let us choose the five points: <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code points = (-5,0), (-2,0), (1,0), (3,-6), (6,2) L11 = [] for point in points : x,y = point L11 += [[x*x*x*x, x*x*x, x*x, x, 1, -y]] print (L11) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ 625.0, -125.0, 25.0, -5.0, 1.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 0.0], # [ 1.0, 1.0, 1.0, 1.0, 1.0, 0.0], # matrix supplied to function solveMbyN() below. [ 81.0, 27.0, 9.0, 3.0, 1.0, 6.0], # 5 rows by 6 columns. [1296.0, 216.0, 36.0, 6.0, 1.0, -2.0]] # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L11) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.02651515151515152, 0.004545454545454542, -0.847727272727273, -0.728787878787879, 1.5454545454545459) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by the 5 points <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> is <math>y = \frac{0.875 x^4 + 0.15 x^3 - 27.975 x^2 - 24.05 x + 51}{33}.</math> {{RoundBoxBottom}} ==Function defined by 3 points and 2 slopes== {{RoundBoxTop|theme=2}} [[File:0314quartic03.png|thumb|400px|'''Figure 1. Quartic function defined by 3 points and 2 slopes.''' </br>Any 3 points on the curve and the slopes at any 2 of these points may be used to define the function.</br></br>Slope at <math>(-2, -2)\ =</math> slope at <math>(6, -4)\ =\ 0.</math>]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 3 unique points on the curve and the slopes at any 2 of these points. For example, let us choose the three points: <math>(-2, -2), (6, -4), (4, 1)</math> At point <math>(-2, -2)</math> slope is <math>0.</math> At point <math>(6, -4)</math> slope is <math>0.</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> Rearrange the standard cubic function of slope to prepare for the calculation of <math>a,b,c,d,e:</math> <math> 4 x^3 a + 3 x^2 b + 2 x c + 1 d + 0 e - s = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code def makeEntry(input) : x,y,s = ( tuple(input) + (None,) )[:3] L1 = [] if s != None : L2 = [ float(v) for v in [4*x*x*x, 3*x*x, 2*x, 1, 0, -s] ] L1 += [ L2 ] L2 = [ float(v) for v in [x*x*x*x, x*x*x, x*x, x, 1, -y]] L1 += [ L2 ] return L1 t1 = ( (-2,-2, 0), # point (-2, -2) with slope 0. (6,-4, 0), # point (6, -4) with slope 0. (4,1), # point (4, 1) ) L1 = [] for v in t1 : L1 += makeEntry ( v ) print (L1) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ -32.0, 12.0, -4.0, 1.0, 0.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 2.0], # [ 864.0, 108.0, 12.0, 1.0, 0.0, 0.0], # matrix supplied to function solveMbyN() below. [1296.0, 216.0, 36.0, 6.0, 1.0, 4.0], # 5 rows by 6 columns. [ 256.0, 64.0, 16.0, 4.0, 1.0, -1.0]]. # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L1) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.03255208333333339, -0.2526041666666665, -0.3072916666666667, 2.84375, 2.375) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by three points and two slopes is: <math>y = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> ===Associated cubic functions=== {{RoundBoxTop|theme=3}} ====When p == -2==== {{RoundBoxTop|theme=1}} [[File:0314_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = -2.''' </br>In this case roots of associated cubic include x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == -2,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 15.25 x^2 + 15.75 x + 105 } {48}.</math> Three blue vertical lines show 3 values of <math>x</math> where <math>g(x) = 0</math> and <math>f(x) = f(-2)</math> In this case roots of <math>g(x)</math> include <math>x = p.</math> {{RoundBoxBottom}} ====When p == 5==== {{RoundBoxTop|theme=1}} [[File:0314_2curves02.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 5.''' </br>In this case the one root of associated cubic excludes x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 5,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 4.3125 x^2 - 36.3125 x - 45.0625 } {48}.</math> Two blue vertical lines show 2 values of <math>x</math> where <math>f(x) = f(5)</math> In this case the one root of <math>g(x)</math> excludes <math>x = p.</math> {{RoundBoxBottom}} ====When p == 6==== {{RoundBoxTop|theme=1}} [[File:0315_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 6.''' </br>In this case the one root of associated cubic includes x = p.]] Quartic function is: <math>y = f(x) </math><math>= \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 6,</math> associated cubic function is: <math>y = g(x) </math><math>= \frac{ 1.5625 x^3 - 2.75 x^2 - 31.25 x - 51 } {48}.</math> One blue vertical line shows 1 value of <math>x</math> where <math>f(x) = f(6)</math> In this case the one root of <math>g(x)</math> includes <math>x = p.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} ==Examples== {{RoundBoxTop|theme=2}} ===Quartic with 2 stationary points=== {{RoundBoxTop|theme=1}} [[File:0315_2statPoints02.png|thumb|400px|'''Graph of quartic function with 2 stationary points.''']] In the diagram the red line represents quartic function <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 2 roots of <math>g(x),\ -1</math> and <math>\frac{-1}{4}</math> show that <math>f(x)</math> has stationary points at <math>x = -1</math> and <math>x = -0.25.</math> {{RoundBoxBottom}} ===Quartic with 1 stationary point=== {{RoundBoxTop|theme=1}} [[File:0315_1statPoint.png|thumb|400px|'''Graph of quartic function with 1 stationary point.''']] In the diagram the red line represents quartic function <math>y = f(x) = \frac{x^4 + 3x^3 + 3x^2 + 3x}{2}</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 1 root of <math>g(x),\ -1.607</math> (approx.), shows that <math>f(x)</math> has 1 stationary point where <math>g(x) = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =First and second derivatives= {{RoundBoxTop|theme=1}} ==Points of inflection== {{RoundBoxTop|theme=2}} [[File:0317_3curves01.png|thumb|400px|'''Graphs of quartic function and first two derivatives.'''</br>Dotted portion of black line shows where f(x) is always concave down.</br>Dotted portion of red line shows where g(x) is decreasing.</br>Dotted portion of blue line shows where h(x) is negative.</br>When h(x) is negative, f(x) is concave down.</br>When h(x) is positive, f(x) is concave up.]] In the diagram the black line has equation: <math>y = f(x) = \frac{x^4 - 16x^3 + 42x^2 + 12x + 4}{144}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{x^3 - 12x^2 + 21x + 3}{36}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{x^2 - 8x + 7}{12}.</math> When <math>x < x_1:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. When <math>x == x_1:</math> * <math>y'</math> is at a local maximum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between up and down. When <math>x_1 < x < x_2:</math> * <math>y'</math> is decreasing. * <math>y''</math> is negative. * <math>f(x)</math> is always concave down. When <math>x == x_2:</math> * <math>y'</math> is at a local minimum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between down and up. When <math>x_2 < x:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. {{RoundBoxTop|theme=1}} The roots of <math>h(x): x_1 = 1, x_2 = 7.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At point <math>p_1</math> concavity of <math>f(x)</math> changes from up to down. At point <math>p_2</math> concavity of <math>f(x)</math> changes from down to up. The points <math>p_1, p_2</math> (the <math>X</math> coordinates of which are roots of <math>h(x)</math>) are the '''points of inflection''' of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Maxima and minima== {{RoundBoxTop|theme=2}} [[File:0317_3curves03.png|thumb|400px|'''Graphs of quartic function and first two derivatives showing maximum and minima.'''</br> Point <math>p_{2}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{2}</math> is concave down. Point <math>p_{2}</math> is local maximum. </br> Point <math>p_{1}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{1}</math> is concave up. Point <math>p_{1}</math> is local minimum. </br> Similarly, point <math>p_{3}</math> is local minimum. ]] In the diagram the black line has equation: <math>y = f(x) = \frac{1.5625x^4 - 12.125x^3 - 14.75x^2 + 136.5x + 114}{48}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{6.25x^3 - 36.375x^2 - 29.5x + 136.5}{48}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{18.75x^2 - 72.75x - 29.5}{48}.</math> Roots of <math>g(x):\ x_1 = -2;\ x_2 = 1.82;\ x_3 = 6.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> At <math>x_1\ h(x_1)</math> is positive. Point <math>p_1</math> is a stationary point and <math>f(x)</math> at <math>p_1</math> is concave up. Point <math>p_1</math> is a local minimum. Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At <math>x_2\ h(x_2)</math> is negative. Point <math>p_2</math> is a stationary point and <math>f(x)</math> at <math>p_2</math> is concave down. Point <math>p_2</math> is a local maximum. Let point <math>p_3</math> on <math>f(x)</math> have coordinates <math>(x_3, f(x_3)).</math> At <math>x_3\ h(x_3)</math> is positive. Point <math>p_3</math> is a stationary point and <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is a local minimum. {{RoundBoxBottom}} ==Quartic with 2 stationary points== {{RoundBoxTop|theme=3}} [[File:0318_3curves01.png|thumb|400px|'''Graph of quartic function with 2 stationary points and first 2 derivatives.''' </br> Black line: <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> </br> Red line: <math>y' = g(x) = 4(4x^3 + 9x^2 + 6x + 1).</math> </br> <math>y'' = h(x) = 4(12x^2 + 18x + 6)</math> </br> Blue line: <math>y'' = 4(2x^2 + 3x + 1) = \frac{h(x)}{6}.</math> </br> Dotted portion of black line shows where <math>f(x)</math> is concave down. ]] In the diagram, point <math>p_1</math> on <math>f(x)</math> has coordinates <math>(x_1, f(x_1)).</math> Similarly, points <math>p_2, p_3</math> have coordinates <math>(x_2, f(x_2)),\ (x_3, f(x_3)).</math> <math>y'</math> has roots: <math>x_1 = -1;\ x_3 = -0.25.</math> Points <math>p_1, p_3</math> are stationary points. <math>y''</math> has roots: <math>x_1 = -1;\ x_2 = -0.5.</math> Points <math>p_1, p_2</math> are points of inflection. At point <math>p_3\ y''</math> is positive. <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is local minimum. {{RoundBoxTop|theme=2}} Summary: * Point <math>p_1</math> is both stationary point and point of inflection. * Point <math>p_2</math> is point of inflection. * Point <math>p_3</math> is both stationary point and local minimum. {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =The simplest quartic function= {{RoundBoxTop|theme=2}} [[File:0320quartic01.png|thumb|400px|'''Graph of simplest quartic function.''' </br> Point <math>p_1\ (-1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_2\ (1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_0\ (0, -1.1^4)</math> is <math>Y</math> intercept of <math>f(x).</math> ]] The simplest quartic function has coefficients <math>b = c = d = 0.</math> Red line in diagram has equation: <math>y = f(x) = x^4 - 1.1^4</math> First derivative (not shown) of <math>f(x):\ y' = g(x) = 4x^3.</math> When <math>x == 0,\ g(x) = 0.</math> There is a stationary point on <math>f(x)</math> when <math>x == 0,</math> point <math>p_0.</math> Second derivative (not shown) of <math>f(x):\ y'' = h(x) = 12x^2.</math> When <math>x == 0,\ h(x) = 0.</math> There is a point of inflection on <math>f(x)</math> when <math>x == 0.</math> For every non-zero value of <math>x,\ h(x)</math> is positive. To left and right of point <math>p_0,\ f(x)</math> is always concave up. Point <math>p_0</math> is both local minimum and absolute minimum. * Point <math>p_0</math> is stationary point and point of inflection and absolute minimum. Curve <math>f(x)</math> is useful for finding the fourth root of a real number. Solve: <math>x = N^{\frac{1}{4}}.</math> <math>x^4 = N.</math> <math>x^4 - N = 0.</math> This is equivalent to finding a root of function <math>y = j(x) = x^4 - N.</math> If you use Newton's method to find a root of <math>j(x),</math> this would be more efficient than solving <math>x = \sqrt{\sqrt{N}}.</math> {{RoundBoxBottom}} =Roots of equal absolute value= {{RoundBoxTop|theme=1}} The standard quartic function: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> For <math>x</math> in <math>(1)</math> substitute <math>(p+q).</math> Call this <math>(2).</math> For <math>x</math> in <math>(1)</math> substitute <math>(p-q).</math> Call this <math>(3).</math> Combine <math>(2)</math> and <math>(3)</math> to eliminate <math>q</math> and produce an equation in <math>p:</math> <math>(- 64aaa)pppppp+</math> <math>(- 96aab)ppppp+</math> <math>(- 32aac - 48abb)pppp+</math> <math>(- 32abc - 8bbb)ppp+</math> <math>(+ 16aae - 4abd - 4acc - 8bbc)pp+</math> <math>(+ 8abe - 2bbd - 2bcc)p+</math> <math>(+ add + bbe - bcd)\ =\ 0\ \dots\ (4).</math> We are interested in coefficient <math>0</math> of <math>(4):\ c_0 = add + bbe - bcd.</math> If <math>c_0 == 0,\ p=0</math> is a solution and function <math>(1)</math> has 2 roots of form <math>0 \pm q</math> where <math>q = \sqrt{\frac{-d}{b}}.</math> An example: {{RoundBoxTop|theme=3}} [[File:0320quartic02.png|thumb|400px|'''Graph of quartic function with 2 roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 12x^3 + 31x^2 + 48x -140}{45}.</math> <math>a,b,c,d,e = 1,-12,31,48,-140</math> <math>c_0 = add + bbe - bcd </math><math>= 1(48)(48) + (-12)(-12)(-140) - (-12)(31)(48) = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-48}{-12}} = \sqrt{4} = \pm 2.</math> The 2 roots of equal absolute value are: <math>2, -2.</math> {{RoundBoxBottom}} The method works with complex roots of equal absolute value: {{RoundBoxTop|theme=2}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with 2 complex roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}.</math> <math>a,b,c,d,e = 1,-3,-1,-27,-90</math> <math>c_0 = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-(-27)}{-3}} = \sqrt{-9} = \pm 3i.</math> The 2 roots of equal absolute value are: <math>3i, -3i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Equal roots= {{RoundBoxTop|theme=2}} Equal roots occur when the function and the slope of the function both equal zero. <math>ax^4 + bx^3 + cx^2 + dx + e = 0\ \dots\ (1)</math> <math>4ax^3 + 3bx^2 + 2cx + d = 0\ \dots\ (2)</math> Begin the process of reducing <math>(1),\ (2)</math> to linear functions. Combine <math>(1),\ (2)</math> to produce 2 cubic functions: <math>Fx^3 + Gx^2 + Hx + J\ \dots\ (1a)</math> where: <math>F = ad;\ G = bd - 4ae;\ H = cd - 3be;\ J = dd - 2ce.</math> <math>fx^3 + gx^2 + hx + j\ \dots\ (2a)</math> where: <math>f = 4a;\ g = 3b;\ h = 2c;\ j = d.</math> Combine <math>(1a),\ (2a)</math> to produce 2 quadratic functions: <math>Kx^2 + Lx + M\ \dots\ (1b)</math> where: <math>K = Gf -Fg;\ L = Hf -Fh;\ M = Jf - Fj.</math> <math>kx^2 + lx + m\ \dots\ (2b)</math> where: <math>k = Fj - Jf;\ l = Gj - Jg;\ m = Hj - Jh.</math> Combine <math>(1b),\ (2b)</math> to produce 2 linear functions: <math>Rx + S\ \dots\ (1c)</math> where: <math>R = Lk - Kl;\ S = Mk - Km.</math> <math>rx + s\ \dots\ (2c)</math> where: <math>r = Km-Mk;\ s = Lm - Ml.</math> From <math>(1c):\ x_1 = \frac{-S}{R}</math> From <math>(2c):\ x_2 = \frac{-s}{r}</math> If <math>x_1 == x_2:</math> <math>\frac{-S}{R} = \frac{-s}{r}</math> <math>Rs = rS</math> <math>Rs - Sr = 0.</math> The value <math>Rs - Sr</math> is in fact: <syntaxhighlight> + 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee - 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd + 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee + 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde + 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee - 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde - 3aabbccdddddd </syntaxhighlight> which, by removing values <math>aa, d</math><math>d</math> (common to all values), may be reduced to: <syntaxhighlight> status = ( + 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee - 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd + 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee + 384abccddde - 54abcddddd + 128acccccee - 80accccdde + 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee - 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde - 3bbccdddd ) </syntaxhighlight> If <math>status == 0,</math> there are at least 2 equal roots which may be calculated as shown below. {{RoundBoxTop|theme=2}} If coefficient <math>d</math> is non-zero, it is not necessary to calculate <math>status.</math> If coefficient <math>d == 0,</math> verify that <math>status = 0</math> before proceeding. {{RoundBoxBottom}} ==No equal roots== {{RoundBoxTop|theme=1}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with no equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}</math> <math>a,b,c,d,e = 1, -3, -1, -27, -90</math> <math>R, S = 15269148, -35977608</math> <math>x_1 = \frac{-S}{R} = \frac{35977608}{15269148} = 2.3562289133617\dots</math> <math>r, s = 35977608, -60634332</math> <math>x_2 = \frac{-s}{r} = \frac{60634332}{35977608} = 1.685335278543253\dots</math> <math>x_1 != x_2.</math> There are no equal roots. {{RoundBoxBottom}} ==Exactly 2 equal roots== {{RoundBoxTop|theme=1}} [[File:0329quartic04.png|thumb|400px|'''Graph of quartic function with exactly 2 equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 + 6x^3 - 48x^2 - 182x + 735}{100}</math> <math>a,b,c,d,e = 1, 6, -48, -182, 735</math> <math>R, S = -1027353600, -7191475200</math> <math>x_1 = \frac{-S}{R} = \frac{7191475200}{-1027353600} = -7</math> <math>r, s = 7191475200, 50340326400</math> <math>x_2 = \frac{-s}{r} = \frac{-50340326400}{7191475200} = -7</math> <math>x_1 = x_2 = -7.</math> There are 2 equal roots at <math>x = -7.</math> {{RoundBoxTop|theme=2}} The following 3 graphs show the steps that lead to calculation of equal roots at point <math>(-7, 0).</math> <gallery> File:0914quartic00.png|<small>Figure 1. graph of <math>f(x)</math> and 2 associated cubic functions.</small> File:0914quartic01.png|<small>Figure 2. graph of <math>f(x)</math> and 2 associated quadratic functions.</small> File:0914quartic02.png|<small>Figure 3. graph of <math>f(x)</math> and 2 associated linear functions.</small> </gallery> In all graphs, all curves have a common root at point <math>(-7, 0).</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=4}} See [[Quartic_function#Function_as_product_of_linear_function_and_cubic | "Function as product of linear function and cubic"]] above. To calculate all roots: <syntaxhighlight lang=python> # python code. a,b,c,d,e = 1,6,-48,-182,735 # The associated cubic: p = -7 A = a B = A*p + b C = B*p + c D = C*P + d # The associated quadratic: a1 = A b1 = a1*p + B c1 = b1*p + C a1,b1,c1 </syntaxhighlight> <syntaxhighlight> (1, -8, 15) </syntaxhighlight> Roots of quadratic function <math>g(x) = x^2 - 8x + 15</math> are <math>3, 5.</math> All roots of <math>f(x)</math> are <math>-7, -7, 3, 5.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Exactly 3 equal roots== {{RoundBoxTop|theme=1}} [[File:0320_2curves01.png|thumb|400px|'''Graph of quartic function with exactly 3 equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 2x^3 - 36x^2 + 162x - 189}{100}</math> <math>a,b,c,d,e = 1, -2, -36, 162, -189</math> <math>R, S = 0, 0\ \dots\dots\ (1c)</math> <math>r, s = 0, 0\ \dots\dots\ (2c)</math> In this case the calculation of <math>x_1, x_2</math> is not appropriate because there are more than 2 equal roots. Try equations <math>(1b), (2b).</math> Both of these are equivalent to: <math>y = g(x) = x^2 - 6x + 9,</math> blue line in diagram. Discriminant of <math>g(x) = (-6)^2 - 4(1)(9) = 0.\ g(x)</math> has two equal roots at <math>x = \frac{-(-6)}{2(1)} = 3.</math> Therefore <math>f(x)</math> has 3 equal roots at <math>x = 3.</math> {{RoundBoxBottom}} ==Four equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves01.png|thumb|400px|'''Graph of quartic function with 4 equal roots and corresponding cubic.''' </br> <math>g(x) = \frac{f'(x)}{4}.</math> ]] Red line in diagram is of function: <math>y = f(x) = x^4 - 20x^3 + 150x^2 - 500x + 625.</math> <math>a,b,c,d,e = 1, -20, 150, -500, 625</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> <math>K, L, M = 0, 0, 0</math> <math>k, l, m = 0, 0, 0</math> In this case <math>(1b), (2b), (1c), (2c)</math> are all null. This is the only case in which <math>(1b), (2b)</math> are null. <math>(1a), (2a)</math> are both equivalent to: <math>y = g(x) = x^3 - 15x^2 + 75x - 125,</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 4 equal roots at <math>x = 5.</math> {{RoundBoxBottom}} ==Two pairs of equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 6x^3 - 11x^2 + 60x + 100}{20}.</math> <math>a,b,c,d,e = 1, -6, -11, 60, 100</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> In this case <math>(1c), (2c)</math> are both null. <math>(1b), (2b)</math> are both equivalent to: <math>y = g(x) = \frac{x^2 - 3x - 10}{20},</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = -2</math> and one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 2 equal roots at <math>x = -2</math> and 2 equal roots at <math>x = 5.</math> {{RoundBoxTop|theme=2}} [[File:0408_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal, complex roots and corresponding quadratic.''' </br> <math>g(x) = (x-(3-2i))(x-(3+2i)) = x^2 - 6x + 13.</math> </br> <math>f(x) = (g(x))^2.</math> ]] This method is valid for complex roots. For example: <math>y = f(x) = x^4 - 12x^3 + 62x^2 - 156x + 169.</math> <math>a,b,c,d,e = 1,-12,62,-156,169.</math> In this case <math>(1c),\ (2c)</math> are both null. <math>(1b),\ (2b)</math> are both equivalent to: <math>y = g(x) = x^2 - 6x + 13,</math> blue line in diagram. Roots of <math>g(x)</math> are <math>3 \pm 2i.</math> <math>f(x)</math> has 2 roots equal to <math>3 + 2i</math> and 2 roots equal to <math>3 - 2i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Summmary== {{RoundBoxTop|theme=1}} {| class="wikitable" |- ! || No equal roots !! 2 equal roots !! 3 equal roots !! 4 equal roots !! 2 pairs of equal roots |- | Cubic: 1(a), 2(a) | different | different | different | same | different |- | Quadratic: 1(b), 2(b) | different | different | same, 1root | null | same, 2roots |- | Linear: 1(c), 2(c) | different | same | null | null | null |} {{RoundBoxBottom}} ==Caution== {{RoundBoxTop|theme=4}} [[File:0915quartic02.png|thumb|400px| Calculation of false equal roots.</br> In this example, method calculates 2 legitimate equal roots at <math>(4, 0)</math> and 2 false equal roots at <math>(7.543296089385474,0).</math> <math></math> ]] Black line in diagram has equation: <math>y = f(x) = 0.012684240362811794x^4</math> <math> -\ 0.19522392290249435x^3 </math> <math> +\ 0.7654478458049887x^2 + 0x - 3. </math> <math>f(x)</math> is a quartic function with exactly 2 equal roots and coefficient <math>d</math> missing. Calculation of equal roots of <math>f(x)</math> gives linear functions <math>(1c), (2c)</math> null and quadratic functions <math>(1b), (2b)</math> with equal roots of <math> (4,0), (7.543296089385474,0). </math> Usually, this indicates that <math>f(x)</math> should have 2 equal roots at <math>(4,0)</math> and 2 equal roots at <math>(7.543296089385474,0).</math> It is obvious that <math>7.543296089385474</math> is not a root of <math>f(x).</math> When <math>x = 7.543296089385474,</math> slope of derivative <math>g(x) = 0.</math> Value of <math>f(x)\ != 0.</math> This example indicates that it would be wise to verify that calculated equal roots are in fact valid roots of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Depressed quartic= {{RoundBoxTop|theme=1}} A depressed quartic is any quartic function with any one or more of coefficients <math>b,c,d</math> missing. Within this section a depressed quartic has coefficient <math>b</math> missing. To produce the depressed quartic: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> <math>y = \frac{(4^4 a^3)(ax^4 + bx^3 + cx^2 + dx + e)}{4^4 a^3}\ \dots\ (2)</math> Let <math>x = \frac{-b + t}{4a}.</math> Substitute in <math>(2),</math> expand and simplify: <math>y = \frac{t^4 + At^2 + Bt + C}{4^4a^3}\ \dots\ (3)</math> where: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> When equated to <math>0,\ (3)</math> becomes the depressed equation: <math>t^4 + At^2 + Bt + C = 0\ \dots\ (4).</math> Be prepared for the possibility that any 1 or more of <math>A, B, C</math> may be zero. ==Coefficient B missing== {{RoundBoxTop|theme=2}} [[File:0322_3curves01.png|thumb|400px|'''Graph of quartic function that resembles a quadratic.''' ]] If coefficient <math>B == 0,\ (4)</math> becomes a quadratic in <math>t^2:</math> <math>t^4 + At^2 + C = 0.</math> <math>(1)</math> has the appearance of a quadratic. The black line: <math>y = f(x) = \frac{x^4 - 4x^3 + 9x^2 - 10x + 5}{10}</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>= 8( 8(-10) - 4(-4)(9) + -64 )</math> <math>= 8(-80 + 144 - 64)</math> <math>= 8(0) = 0.</math> The red line: <math>y' = g(x) = \frac{4x^3 - 12x^2 + 18x - 10}{10}</math> <math>y' = g(x) = 0</math> where <math>x = 1.</math> The grey line: <math>y'' = h(x) = \frac{12x^2 - 24x + 18}{10}</math> * Absolute minima of <math>f(x)</math> and of <math>h(x)</math> and point of inflection of <math>g(x)</math> occur where <math>x = \frac{-b}{4} = 1.</math> * <math>y''</math> is always positive. <math>f(x)</math> is always concave up. * <math>f(1 + p) = f(1-p).</math> {{RoundBoxTop|theme=3}} If <math>(1)</math> contains 2 pairs of equal roots, coefficient <math>B = 0.</math> The converse is not necessarily true. If <math>(1)</math> contains 4 equal roots, coefficients <math>A = B = C = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Coefficient C missing== {{RoundBoxTop|theme=2}} [[File:0501quartic01.png|thumb|400px|'''Graph of quartic function with coefficient C of depressed function missing.''' </br> <math>Y</math> axis compressed for clarity. ]] If coefficient <math>C == 0,\ (4)</math> becomes: <math>t^4 + At^2 + Bt = t(t^3 + At + B) = 0</math> in which case <math>t = 0</math> is a solution and <math>x = \frac{-b}{4a}</math> is a root of <math>(1).</math> Curve (red line) in example has equation: <math>y = f(x) = 8x^4 + 16x^3 + 24x^2 + 89x + 40.</math> Coefficients of depressed function are: <syntaxhighlight lang=python> # python code a,b,c,d,e = 8, 16, 24, 89, 40 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b A,B,C </syntaxhighlight> <syntaxhighlight> (1536, 299008, 0) </syntaxhighlight> Coefficient <math>C</math> of depressed function is missing. <math>t = 0</math> is a solution. Using <math>x = \frac{-b + t}{4a},</math> one root of <math>f(x) = \frac{-16 + 0}{4(8)} = -0.5.</math> {{RoundBoxBottom}} ==Resolvent cubic== {{RoundBoxTop|theme=1}} This section introduces a special cubic function called "resolvent" because it helps to resolve a requirement, the calculation of the roots of the quartic. The depressed quartic: <math>t^4 + At^2 + Bt + C\ \dots\ (1)</math> For <math>t</math> substitute <math>(u+v)\ \dots\ (2)</math> For <math>t</math> substitute <math>(u-v)\ \dots\ (3)</math> <math>(2)+(3):\ 2Auu + 2Avv + 2Bu + 2C + 2uuuu + 12uuvv + 2vvvv\ \dots\ (4)</math> Simplify <math>(4):\ Auu + AV + Bu + C + uuuu + 6uuV + VV\ \dots\ (4a)</math> <math>(2)-(3):\ 4Auv + 2Bv + 8uuuv + 8uvvv\ \dots\ (5)</math> Simplify <math>(5):\ 2Au + B + 4uuu + 4uV\ \dots\ (5a)</math> From <math>(5a):\ 4uV = -(2Au + B + 4uuu)\ \dots\ (5b)</math> <math>(4a)*4u4u:\ 4u4uAuu + A4u(4uV) + 4u4uBu + 4u4uC + 4u4uuuuu + 6uu4u(4uV) + (4uV)(4uV)\ \dots\ (6)</math> In <math>(6)</math> replace <math>4uV</math> with <math>(-(2Au + B + 4uuu)),</math> expand, simplify, gather like terms and result is: <math>Pu^6 + Qu^4 + Ru^2 + S</math> or {{RoundBoxTop|theme=2}} <math>PU^3 + QU^2 + RU + S\ \dots\ (7)</math> where: <math>U = u^2</math> <math>P = 64</math> <math>Q = 32A</math> <math>R = 4A^2 - 16C</math> <math>S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} From <math>(5b):\ V = v^2 = \frac{-(2Au + B + 4uuu)}{4u} = -(\frac{A}{2} + U) - \frac{B}{4u}\ \dots\ (8)</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Some simple changes reduce the number of calculations and also the sizes of coefficients <math>P, Q, R, S.</math> <math>A2 = 8ac - 3b^2</math> where <math>A2 = \frac{A}{2}</math> <math>B4 = 16a^2 d - 8abc + 2b^3</math> where <math>B4 = \frac{B}{4}</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> Then: <math>P = 64</math> <math>Q = 32(A2)(2) = 64A2</math> <math>R = 4(A2)(2)(A2)(2) - 16C = 16A2^2 - 16C</math> <math>S = -(B4)(4)(B4)(4) = -16 B4^2</math> Divide all 4 coefficients by <math>16:</math> <math>P = 4</math> <math>Q = 4A2</math> <math>R = A2^2 - C</math> <math>S = -B4^2</math> <math>V = v^2 = -(A2 + U) - \frac{B4}{u}.</math> A close examination of coefficients <math>R, S</math> shows that both coefficients are always exactly divisible by <math>4.</math> Therefore, all four coefficients may be defined as follows: <math>P = 1</math> <math>Q = A2</math> <math>R = \frac{A2^2 - C}{4}</math> <math>S = \frac{-B4^2}{4}</math> Actual calculations of values <math>R, S</math> are: <syntaxhighlight lang=python> # python code. R = ( (16)*(a)*(a)*(b)*(d) + (16)*(a)*(a)*(c)*(c) + (3)*(b)*(b)*(b)*(b) + (-16)*(a)*(b)*(b)*(c) + (-64)*(a)*(a)*(a)*(e) ) S = ( (64)*(a)*(a)*(a)*(b)*(c)*(d) + (8)*(a)*(b)*(b)*(b)*(b)*(c) + (-16)*(a)*(a)*(b)*(b)*(b)*(d) + (-16)*(a)*(a)*(b)*(b)*(c)*(c) + (-1)*(b)*(b)*(b)*(b)*(b)*(b) + (-64)*(a)*(a)*(a)*(a)*(d)*(d) ) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =Solving quartic equation= This section presents 4 examples that show how to use the depressed quartic and the resolvent cubic to solve the quartic equation. ==Four real roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves01.png|thumb|400px|'''Graphs of quartic function with 4 real roots and associated resolvent cubic.''' </br>Resolvent cubic contains 3 real, positive roots. ]] Calculate roots of: <math>y = f(x) = x^4 - x^3 - 19x^2 - 11x + 30</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,-1,-19,-11,30 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -310 -1320 6669 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 -9920 277696 -1742400 </syntaxhighlight> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = 64x^3 - 9920x^2 + 277696x - 1742400.</math> There are 3 real, positive roots: <math>9, 25, 121.</math> Using 3 roots of <math>g(x),</math> calculate 4 roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code for U in (9, 25, 121) : print ('\nU =', U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** .5 for t in (u+v, u-v) : x = (-b+t) / (4*a) y = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x:',x, '; y:',y) </syntaxhighlight> <syntaxhighlight> U = 9 x: 5.0 ; y: 0.0 x: -3.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 U = 25 x: 5.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -3.0 ; y: 0.0 U = 121 x: 5.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: -3.0 ; y: 0.0 </syntaxhighlight> Roots of <math>f(x)</math> are: <math>5, 1, -2, -3.</math> {{RoundBoxBottom}} All 3 values of <math>U</math> produce the same results, but not in same sequence. It is not necessary to calculate all 3 roots of resolvent cubic. Any one non-zero root is sufficient to do the job. ==Two real and two complex roots== ===Example 1=== {{RoundBoxTop|theme=4}} [[File:0323_2curves02.png|thumb|400px|'''Graphs of quartic function with 2 real and 2 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 + 2x^3 + 18x^2 - 70x - 87</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,2,18,-70,-87 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> 264 -5568 -12208 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 8448 474112 -31002624 </syntaxhighlight> {{RoundBoxBottom}} Calculate one real root of cubic function: <math>y = g(x) = 64x^3 + 8448x^2 + 474112x -31002624.</math> <math>36</math> is one real root. Choose <math>U = 36.</math> Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 36 u1 = U**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) if V >= 0 : v = V**.5 else : v = 1j * (-V)**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =',x, '; sum =',sum ) </syntaxhighlight> <syntaxhighlight> x = 3.0 ; sum = 0.0 x = -1.0 ; sum = 0.0 x = (-2+5j) ; sum = 0j x = (-2-5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=4}} [[File:0321_2curves00.png|thumb|400px| Coefficient <math>d</math> of <math>g(x) = 0.</math> <math></math> ]] Calculate roots of: <math>y = f(x)</math> <math>= 3x^4 - 6x^3 - 41x^2 +44x - 189</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 3, -6, -41, 44, -189 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -2184, 0, -1229040 </syntaxhighlight> Notice that coefficient <math>B = 0.</math> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 1, -1092, 605376, 0 </syntaxhighlight> Notice that coefficient <math>S = 0.</math> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = x^3 - 1092x^2 + 605376x + 0.</math> Roots are <math>0, 546 \pm 554.3103823671355j.</math> Value <math>0</math> cannot be used because it will cause error <code>Divide by zero</code> at statement <code>V = -(A/2 + U) - B/(4*u)</code>. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 546+554.3103823671355j print ('\nU =',U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** 0.5 s1 = '\nu,v' print (s1,eval(s1)) for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> U = (546+554.3103823671355j) u,v ((25.729935131257832+10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (4.788322521876306 + 0j) ; sum = (1.9895196601282805e-13 + 0j) x = (0.5 + 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) u,v ((-25.729935131257832-10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (0.5 - 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) x = (-3.7883225218763052 + 0j) ; sum = (1.7053025658242404e-13 + 0j) </syntaxhighlight> Values of <math>x</math> are: <math>-3.7883225218763052, 4.788322521876306, 0.5 \pm 1.795283650280614j</math> <math></math><math></math> {{RoundBoxBottom}} ====Depressed quartic as quadratic==== In this example coefficient <math>B</math> of depressed quartic <math>= 0.</math> Therefore, resolvent cubic can be ignored and depressed quartic processed as quadratic in <math>T = t^2.</math> <math>t^4 - 2184t^2 + (0)t - 1229040</math> <math>T^2 - 2184T - 1229040</math> where <math>T = t^2.</math> Solutions of this quadratic are: <math>T_1, T_2 = 2648.1182474349434, -464.11824743494344</math> <syntaxhighlight lang=python> T1,T2 = 2648.1182474349434, -464.11824743494344 t1 = T1 ** 0.5; t2 = ((-T2) ** 0.5) * 1j for t in (t1,-t1,t2,-t2) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> x = 4.788322521876305 ; sum = -1.7053025658242404e-13 x = -3.788322521876305 ; sum = -1.7053025658242404e-13 x = (0.5 + 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) x = (0.5 - 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) </syntaxhighlight> or <syntaxhighlight> x = 0.5 ± 4.288322521876305, 0.5 ± 1.7952836502806138j </syntaxhighlight> With precision of 15, values of <math>x</math> are same as those shown above. When roots of quartic function are of form <code>p ± q, p ± r,</code> coefficient <math>B</math> of depressed function <math>= 0.</math> ==Four complex roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves03.png|thumb|400px|'''Graphs of quartic function with 4 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 - 20x^3 + 408x^2 + 2296x + 18020</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 4128 344064 9683200 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 64 132096 -86769664 -118380036096 </syntaxhighlight> {{RoundBoxBottom}} Calculate one root of cubic function: <math>y = g(x) = 64x^3 + 132096x^2 - 86769664x - 118380036096.</math> There are 3 real roots: <math>-2304, -784, 1024.</math> Choose <math>U = -784.</math> Negative <math>U</math> is chosen here to show that any 1 of the roots produces the correct result. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = -784 u1 = 1j * (-U)**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) v = V**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> # python expresses complex numbers with 'j'. x = (13+19j) ; sum = 0j x = (-3-5j) ; sum = 0j x = (13-19j) ; sum = 0j x = (-3+5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} =Quartic formula= The substitutions made above can be used to produce a formula for the solution of the quartic equation. See main articles [https://en.wikipedia.org/wiki/Quartic_equation#The_general_case "The general case" ] or [https://en.wikipedia.org/wiki/Quartic_function#General_formula_for_roots "General formula for roots."] {{RoundBoxTop|theme=2}} Both links above point to formula for equation <math>x^4 + ax^3 + bx^2 + cx + d = 0.</math> {{RoundBoxBottom}} Given quartic equation: <math>ax^4 + bx^3 + cx^2 + dx + e = 0,</math> calculate the 4 values of <math>x.</math> <math>x = \frac{-b + t}{4a}</math> where: {{RoundBoxTop|theme=2}} Coefficients of depressed quartic: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of resolvent cubic: <math>a_1 = P = 64</math> <math>b_1 = Q = 32A</math> <math>c_1 = R = 4A^2 - 16C</math> <math>d_1 = S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of depressed cubic: <math>A_1 = 9a_1 c_1 - 3b_1^2</math> <math>B_1 = 27a_1^2d_1 - 9a_1b_1c_1 + 2b_1^3</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of resolvent cubic: <math>C_1 = \frac{-A_1}{3} = b_1^2 - 3a_1 c_1</math> <math>\Delta = B_1^2 - 4C_1^3\ \dots\dots\ \Delta</math> may be negative. <math>\delta = \sqrt{\Delta}</math> <math>W = \frac{-B_1 + \delta}{2}</math> <math>w = \sqrt[3]{W}</math> <math>t_1 = w + \frac{C_1}{w}</math> <math>U = \frac{-b_1 + t_1}{3a_1}</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of quartic: <math>u = \sqrt{U}\ \dots\dots\ u</math> may be positive or negative. <math>V = -(\frac{A}{2} + U) - \frac{B}{4u}</math> <math>v = \sqrt{V}\ \dots\dots\ v</math> may be positive or negative. <math>t = u + v</math> {{RoundBoxBottom}} Formula above produces one value of <math>x.</math> Python code below utilizes <math>\pm \sqrt{U}</math> and <math>\pm \sqrt{V}</math> to produce 4 values of <math>t</math> and then, four values of <math>x.</math> ==An example:== {{RoundBoxTop|theme=4}} [[File:0330quartic01.png|thumb|400px|'''Graph of quartic function with 2 real roots.'''</br><math>Y</math> axis compressed for clarity.]] Calculate roots of <math>f(x) = 4x^4 + 4x^3 - 75x^2 - 776x - 1869.</math> <syntaxhighlight lang=python> # Python code. a,b,c,d,e = 4, 4, -75, -776, -1869 values_of_t = [ t # Coefficients of depressed quartic: for A in (16*a*c - 6*b*b,) for B in (64*a*a*d -32*a*b*c + 8*b*b*b,) for C in (256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b,) # Coefficients of resolvent cubic: for a1 in (64,) for b1 in (32*A,) for c1 in (4*A*A - 16*C,) for d1 in (-B*B,) for U in [ # The resolvent cubic: (-b1+t1)/(3*a1) # Coefficients of depressed resolvent cubic: for A1 in (9*a1*c1 - 3*b1*b1,) for B1 in (27*a1*a1*d1 - 9*a1*b1*c1 + 2*b1*b1*b1,) # One root of resolvent cubic: for C1 in (-A1/3,) for Δ in (B1*B1 - 4*C1*C1*C1,) for δ in (Δ**0.5,) for W in ((-B1 + δ)/2,) for w in (W**(1/3),) for t1 in (w + C1/w,) # See note below. ] # Prepare to calculate 4 values of t. for u1 in (U**.5,) for v1 in ( -(A/2 + U) ,) # Calculate 4 values of t. for u in (u1, -u1,) for V in ( v1 - B/(4*u),) for v in (V**.5,) for t in (u+v, u-v) ] print ('values_of_t =', values_of_t) </syntaxhighlight> {{RoundBoxTop|theme=4}} <syntaxhighlight> values_of_t = [116, -44, (-36+64j), (-36-64j)] </syntaxhighlight> Because <math>f(t)</math> is a depressed quartic function, sum of four <code>values_of_t</code> <math>= 116 - 44 - 36(2) = 0.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> # Python code. # Calculate 4 separate roots. values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] print ('values_of_x =', values_of_x) </syntaxhighlight> <syntaxhighlight lang=python> values_of_x = [7, -3, (-2.5 + 4j), (-2.5 - 4j)] </syntaxhighlight> In python the imaginary part of a complex number is shown with <math>j</math> instead of <math>i.</math> If <math>A == B == C == 0,\ f(x)</math> contains 4 equal roots and <math>x = \frac{-b}{4a}.</math> If <math>f(x)</math> contains 3 or more equal roots, statement <code>for t1 in (w + C1/w,)</code> fails with divisor <math>w = 0.</math> Before using this formula, check for equal roots as in [[Quartic_function#Exactly_3_equal_roots | "Exactly 3 equal roots"]] above. {{RoundBoxTop|theme=1}} Values displayed above have been edited slightly. Actual calculated values were: <syntaxhighlight> values_of_x = [7.000000000000001, -3.0000000000000044, (-2.499999999999998+4.000000000000001j), (-2.4999999999999987-4.000000000000001j)] </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ==In practice== {{RoundBoxTop|theme=5}} The following Python code implements the quartic formula. However, under statement <code>if B4 == 0 :</code> there is code that processes the depressed quartic as a quadratic in <math>T = t^2.</math> This ensures that execution of formula does not fail with error <code>Divide by zero</code> at statement <code>for V in ( v1 - B4/u,).</code> {{RoundBoxTop|theme=4}} <syntaxhighlight lang=python> # python code import cmath cxSqrt = cmath.sqrt # Square root of complex number. def rootsOfQuartic (abcde) : ''' x1,x2,x3,x4 = rootsOfQuartic ((a,b,c,d,e)) Each member of input must be int or float or Decimal object. Int or Decimal object in input is quietly converted to float. Output may be None. ''' def formatResults (x1x2x3x4) : ''' This function improves appearance of results. (8 + 0j) becomes 8.0 ''' values_of_x = list (x1x2x3x4) for p in (0,1,2,3) : v = values_of_x[p] if isinstance (v, complex) and (v.imag == 0) : values_of_x[p] = v.real return values_of_x status = 0 try : a,b,c,d,e = [ float(v) for v in abcde ] except : status = 1 if status : print ('rootsOfQuartic () 1: Error creating coefficients a,b,c,d,e.') return None if a == 0 : print ('rootsOfQuartic () 2: Coefficient a must be non-zero.') return None # Coefficients of depressed quartic, modified. # A = 16*a*c - 6*b*b A2 = 8*a*c - 3*b*b # B = 64*a*a*d - 32*a*b*c + 8*b*b*b B4 = 16*a*a*d - 8*a*b*c + 2*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b if B4 == 0 : # B = 0. # Result returned from this section is type tuple, indicating that coefficient B4 = 0. if A2==C==0 : # 4 equal roots. root = -b/(4*a) return tuple(formatResults((root,root,root,root))) # t**4 + At**2 + (0)t + C # Depressed quartic is quadratic in T: # T**2 + AT + C where T = t**2 # T**2 + 2(A2)T + C where A = 2(A2) # -2(A2) +/- (4(A2)(A2) - 4C)**0.5 # T = -------------------------------- = -A2 +/- ((A2)(A2) - C)**0.5 # 2 disc = A2*A2 - C if disc >= 0 : root = disc ** 0.5 else : root = ((-disc) ** 0.5) * 1j T1 = -A2 - root ; T2 = -A2 + root t1 = cxSqrt(T1) ; t2 = cxSqrt(T2) values_of_t = (t1,-t1,t2,-t2) values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] return tuple(formatResults(values_of_x)) # B4 is non-zero. Therefore, all of (S, U, u) are non-zero. P,Q,R,S = 1, A2, (A2*A2 - C)/4, -B4*B4/4 # str1 = 'P,Q,R,S' ; print (str1, eval(str1)) U = oneRootOfCubic((P,Q,R,S)) # Resolvent cubic. if U > 0 : sqrtU = U ** 0.5 elif U == 0 : # This should not happen. print ('rootsOfQuartic () 3: Internal error.') return None else : sqrtU = ((-U) ** 0.5) * 1j v1 = -(A2+U) values_of_t = [ t for u in (sqrtU, -sqrtU) for V in ( v1 - B4/u, ) for v in ( cxSqrt(V), ) for t in (u+v, u-v) ] values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] # Result returned from this section is type list, indicating that coefficient B4 != 0. return formatResults(values_of_x) </syntaxhighlight> For function <code>oneRootOfCubic()</code> see [[Cubic_function#In_practice | Cubic_function: In_practice.]] {{RoundBoxBottom}} ===Examples=== {{RoundBoxTop|theme=2}} Python function <code>equalRoots()</code> below implements <code>status</code> as presented under [https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above. <syntaxhighlight lang=python> # python code def equalRoots(abcde) : ''' This function returns True if quartic function contains at least 2 equal roots. ''' a,b,c,d,e = abcde aa = a*a ; aaa = aa*a bb = b*b ; bbb = bb*b ; bbbb = bb*bb cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd ee = e*e ; eee = ee*e ; eeee = ee*ee v1 = ( +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd +20*bb*ccc*dd*e ) v2 = ( -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd ) return (v1+v2) == 0 t1 = ( ((1, -1, -19, -11, 30), '4 unique, real roots.'), ((4, 4,-119, -60, 675), '4 unique, real roots, B4 = 0.'), ((1, 6, -48,-182, 735), '2 equal roots.'), ((1,-12, 50, -84, 45), '2 equal roots. B4 = 0.'), ((1,-20, 146,-476, 637), '2 equal roots, 2 complex roots.'), ((1,-12, 58,-132, 117), '2 equal roots, 2 complex roots. B4 = 0.'), ((1, -2, -36, 162, -189), '3 equal roots.'), ((1,-20, 150,-500, 625), '4 equal roots. B4 = 0.'), ((1, -6, -11, 60, 100), '2 pairs of equal roots, B4 = 0.'), ((4, 4, -75,-776,-1869), '2 complex roots.'), ((1,-12, 33, 18, -208), '2 complex roots, B4 = 0.'), ((1,-20, 408,2296,18020), '4 complex roots.'), ((1,-12, 83, -282, 442), '4 complex roots, B4 = 0.'), ((1,-12, 62,-156, 169), '2 pairs of equal complex roots, B4 = 0.'), ) for (abcde, comment) in t1 : print () fourRoots = rootsOfQuartic (abcde) print (comment) print (' Coefficients =', abcde) print (' Four roots =', fourRoots) print (' Equal roots detected:', equalRoots(abcde)) # Check results. a,b,c,d,e = abcde for x in fourRoots : # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0 sum = (a*x**4 + b*x**3 + c*x**2 + d*x + e) if sum : # Create exception. 1/0 </syntaxhighlight> <syntaxhighlight> 4 unique, real roots. Coefficients = (1, -1, -19, -11, 30) Four roots = [5.0, 1.0, -2.0, -3.0] Equal roots detected: False 4 unique, real roots, B4 = 0. Coefficients = (4, 4, -119, -60, 675) Four roots = (2.5, -3.0, 4.5, -5.0) Equal roots detected: False 2 equal roots. Coefficients = (1, 6, -48, -182, 735) Four roots = [5.0, 3.0, -7.0, -7.0] Equal roots detected: True 2 equal roots. B4 = 0. Coefficients = (1, -12, 50, -84, 45) Four roots = (3.0, 3.0, 5.0, 1.0) Equal roots detected: True 2 equal roots, 2 complex roots. Coefficients = (1, -20, 146, -476, 637) Four roots = [7.0, 7.0, (3+2j), (3-2j)] Equal roots detected: True 2 equal roots, 2 complex roots. B4 = 0. Coefficients = (1, -12, 58, -132, 117) Four roots = ((3+2j), (3-2j), 3.0, 3.0) Equal roots detected: True 3 equal roots. Coefficients = (1, -2, -36, 162, -189) Four roots = [3.0, 3.0, 3.0, -7.0] Equal roots detected: True 4 equal roots. B4 = 0. Coefficients = (1, -20, 150, -500, 625) Four roots = (5.0, 5.0, 5.0, 5.0) Equal roots detected: True 2 pairs of equal roots, B4 = 0. Coefficients = (1, -6, -11, 60, 100) Four roots = (5.0, -2.0, 5.0, -2.0) Equal roots detected: True 2 complex roots. Coefficients = (4, 4, -75, -776, -1869) Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)] Equal roots detected: False 2 complex roots, B4 = 0. Coefficients = (1, -12, 33, 18, -208) Four roots = ((3+2j), (3-2j), 8.0, -2.0) Equal roots detected: False 4 complex roots. Coefficients = (1, -20, 408, 2296, 18020) Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)] Equal roots detected: False 4 complex roots, B4 = 0. Coefficients = (1, -12, 83, -282, 442) Four roots = ((3+5j), (3-5j), (3+2j), (3-2j)) Equal roots detected: False 2 pairs of equal complex roots, B4 = 0. Coefficients = (1, -12, 62, -156, 169) Four roots = ((3+2j), (3-2j), (3+2j), (3-2j)) Equal roots detected: True </syntaxhighlight> When description contains note <math>B4 = 0,</math> depressed quartic was processed as quadratic in <math>t^2.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Two Conic Sections= Examples of conic sections include: ellipse, circle, parabola and hyperbola. This section presents examples of two conic sections, circle and ellipse, and how to calculate the coordinates of the point/s of intersection, if any, of the two sections. Let one section with name <math>ABCDEF</math> have equation <math>Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.</math> Let other section with name <math>abcdef</math> have equation <math>ax^2 + by^2 + cxy + dx + ey + f = 0.</math> Because there can be as many as 4 points of intersection, a special "resolvent" quartic function is used to calculate the <math>x</math> coordinates of the point/s of intersection. Coefficients of associated "resolvent" quartic are calculated as follows: <syntaxhighlight lang=python> # python code def intersection_of_2_conic_sections (abcdef, ABCDEF) : ''' A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF) where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function: y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_ ''' A,B,C,D,E,F = ABCDEF a,b,c,d,e,f = abcdef G = ((-1)*(B)*(a) + (1)*(A)*(b)) H = ((-1)*(B)*(d) + (1)*(D)*(b)) I = ((-1)*(B)*(f) + (1)*(F)*(b)) J = ((-1)*(C)*(a) + (1)*(A)*(c)) K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c)) L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c)) M = ((-1)*(E)*(f) + (1)*(F)*(e)) g = ((-1)*(C)*(b) + (1)*(B)*(c)) h = ((-1)*(E)*(b) + (1)*(B)*(e)) A_ = (J)*(g) + (G)*(G) B_ = (J)*(h) + (K)*(g) + (2)*(G)*(H) C_ = (K)*(h) + (L)*(g) + (2)*(G)*(I) + (H)*(H) D_ = (L)*(h) + (M)*(g) + (2)*(H)*(I) E_ = (M)*(h) + (I)*(I) str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_) print (str1) return A_,B_,C_,D_,E_ </syntaxhighlight> == With no common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections06.png|thumb|400px|'''Two conic sections with no common point.''' </br> Resolvent quartic function (black curve) has no real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 80 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.98x^2 + 6850.532x - 22836.7009.</math> <math>f(x)</math> has no real roots. Therefore, there is no point of intersection. {{RoundBoxBottom}} ==With one common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections05.png|thumb|400px|'''Two conic sections with one common point.''' </br> Resolvent quartic function (black curve) has two equal, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 6.8x - 17.6y + 73 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.84x^2 + 7456.48x - 24355.36.</math> Roots of <math>f(x)</math> are: <math>(-14-22.978250586152114j),</math> <math>(-14+22.978250586152114j),</math> <math>5.8, 5.8.</math> <math>f(x)</math> has 2 equal, real roots at <math>x = 5.8,</math> effectively 1 real root where <math>x = 5.8</math> Therefore, there is one point of intersection where <math>x = 5.8.</math> {{RoundBoxBottom}} ==With two common points== ===Example 1=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections04.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two unique, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 6.8x - 17.6y + 64 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.66x^2 + 8235.556x - 26681.1841.</math> Roots of <math>f(x)</math> are: <math>(-14.361578825892241-23.341853011785357j),</math> <math>(-14.361578825892241+23.341853011785357j),</math> <math>4.59885619413921, 7.72430145764527.</math> <math>f(x)</math> has 2 unique, real roots at <math>x = 4.59885619413921, 7.72430145764527.</math> Therefore, there are two points of intersections where <math>x = 4.59885619413921, 7.72430145764527.</math> {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections01.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two pairs of equal roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 18.8x - 1.6y + 53 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 + 37.6x^3 - 504.24x^2 + 2835.04x - 5685.16.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 13, 13.</math> <math>f(x)</math> has 2 pairs of equal roots at <math>x = 5.8, 13,</math> effectively 2 real roots. Therefore, there are two points of intersection where <math>x = 5.8, 13.</math> {{RoundBoxBottom}} ==With 3 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections03.png|thumb|400px|'''Two conic sections with three common points.''' </br> Resolvent quartic function (black curve) has one pair of equal roots and 2 unique, real roots. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 17.6x - 3.2y + 55 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 6.83589838486224, 13.7641016151377 .</math> <math>f(x)</math> has 1 pair of equal roots at <math>x = 5.8</math> and 2 unique, real roots at <math>x = 6.83589838486224, 13.7641016151377,</math> effectively 3 real roots. Therefore, there are three points of intersection where <math>x = 5.8, 6.83589838486224, 13.7641016151377.</math> {{RoundBoxBottom}} ==With 4 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections02.png|thumb|400px|'''Two conic sections with four common points.''' </br> Resolvent quartic function (black curve) has 4 unique, real roots. </br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 18.8x - 1.6y + 62.99 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x)</math><math>= -x^4 + 37.6x^3 - 504.4398x^2 + 2838.79624x - 5544.61147921.</math> Roots of <math>f(x)</math> are:<math>4.36661032156283,</math><math>8.77936456353008,</math> <math>10.0206354364699,</math><math>14.4333896784371.</math> <math>f(x)</math> has 4 real roots as shown above. Therefore, there are four points of intersection where <math>x = 4.36661032156283,</math> <math>8.77936456353008,</math><math>10.0206354364699,</math><math>14.4333896784371.</math> {{RoundBoxBottom}} ===Values of y=== Section above calculates <math>x</math> coordinates of the four common points. This section calculates <math>y</math> coordinates. Here is the lazy way to do it: {{RoundBoxTop|theme=2}} [[File:0818ResQuartic.png|thumb|400px|'''Graphs of 2 curves, red and blue, with coefficients of <math>x,y</math> reversed.''' </br> Roots of "resolvent" quartic, black curve, are interpreted as values of <math>y.</math> ]] <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 # Coefficients of x,y are reversed. abcde = intersection_of_2_conic_sections ((b,a,0,e,d,f), (B,A,C,E,D,F)) values_of_y = rootsOfQuartic (abcde) print (values_of_y) </syntaxhighlight> <syntaxhighlight> [1.6215767432213488, -0.021576743221348723, 5.862095579401657, -4.262095579401658] </syntaxhighlight> By inspection of the diagram above, it is obvious that: <syntaxhighlight lang=python> # python code four_common_points = ( (4.36661032156283, 1.6215767432213488), (8.77936456353009, -4.262095579401658), (10.0206354364699, 5.862095579401657), (14.4333896784372, -0.021576743221348723), ) </syntaxhighlight> Check results: <syntaxhighlight lang=python> # python code def sum_zero (input) : ''' This function calculates sum of all values in input. If sum is very close to 0, this function returns 0. ''' sum = 0 for v in input : sum += v if sum == 0 : return sum tolerance = 1e-14 max = sorted([ abs(v) for v in input ])[-1] if abs(sum) < tolerance*max : return 0 return sum # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 for (x,y) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum1 = sum_zero(values) if sum1 : print ('Error: sum1 =',sum1) values = a*x**2, b*y**2, d*x, e*y, f sum2 = sum_zero(values) if sum2 : print ('Error: sum2 =',sum2) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. Check results with coefficients of <math>x, y</math> reversed: <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = B,A,C,E,D,F # The circle: a,b,c,d,e,f = b,a,c,e,d,f for (y,x) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum3 = sum_zero(values) if sum3 : print ('Error: sum3 =',sum3) values = a*x**2, b*y**2, d*x, e*y, f sum4 = sum_zero(values) if sum4 : print ('Error: sum4 =',sum4) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. {{RoundBoxBottom}} =Links to related topics= {{RoundBoxTop|theme=1}} [[Cubic_function#Cubic_formula | "Cubic formula"]] [[Complex_square_root | "Complex square root" ]] {{RoundBoxBottom}} [[Category:Mathematics]] 3rydcgqezv2eayvt4rs9zkatd182aw0 2844178 2844177 2026-11-03T11:24:38Z ThaniosAkro 2805358 /* With one common point */ 2844178 wikitext text/x-wiki The quartic function is the bridge between the cubic function and more advanced functions such as the quintic and sextic. =Objective= {{RoundBoxTop|theme=3}} [[file:Books-aj.svg_aj_ashton_01f.png|right|100px]] * Present quartic function and quartic equation. * Introduce the concept of roots of equal absolute value. * Show how to predict and calculate equal roots, techniques that will be useful when applied to higher order functions. * Simplify the depressed quartic. * Show that the quartic equation is effectively solved when at least one root is known. * Present the "resolvent" cubic function. * Show how to derive and use the quartic formula. {{RoundBoxBottom}} =Lesson= {{RoundBoxTop|theme=2}} ==Introduction== The quartic function is the sum of powers of <math>x</math> from <math>0</math> through <math>4</math>: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx^1 + ex^0</math> usually written as: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx + e.</math> If <math>e == 0</math> the function becomes <math>x(ax^3 + bx^2 + cx + d).</math> Within this page we'll say that: * both coefficients <math>a, e</math> must be non-zero, * coefficient <math>a</math> must be positive (simply for our convenience), * all coefficients must be real numbers, accepting that the function may contain complex roots. The quartic equation is the quartic function equated to zero: <math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>. Roots of the function are values of <math>x</math> that satisfy the quartic equation. {{RoundBoxTop|theme=1}} Because the function is "quartic" (maximum power of <math>x</math> is <math>4</math>), the function contains exactly <math>4</math> roots, an even number of complex roots and an even number of real roots. Other combinations of real and complex roots are possible, but they produce complex coefficients. {{RoundBoxBottom}} {{RoundBoxTop|theme=7}} [[File:0312quartic01a.png|thumb|400px|'''Graph of typical quartic function showing minima and maximum.''']] The figure shows a typical quartic function. The function crosses the <math>X</math> axis in 4 different places. The function has 4 roots: <math>(-2,0), (1,0), (5,0), (10,0).</math> This function contains one local minimum, one local maximum and one absolute minimum. There is no absolute maximum. Because the function contains one absolute minimum: * If abs(<math>x</math>) is very large, <math>f(x)</math> is always positive. * If absolute minimum is above <math>X</math> axis, curve does not cross <math>X</math> axis and function contains only complex roots. * There is always at least one point where the curve is parallel to <math>X</math> axis. The curve is never parallel to the <math>Y</math> axis. For any real value of <math>x</math> there is always a real value of <math>y.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0312quartic02.png|thumb|400px|'''When coefficient d is missing, there is a stationary point at x = 0.''']] If coefficient <math>d</math> is missing, the quartic function becomes <math>y = ax^4 + bx^3 + cx^2 + e,</math> and <math>y' = 4ax^3 + 3bx^2 + 2cx = x(4ax^2 + 3bx + 2c).</math> For a stationary point <math>y' = x(4ax^2 + 3bx + 2c) = 0.</math> When coefficient <math>d</math> is missing, there is always a stationary point at <math>x = 0.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} [[File:0319quartic02.png|thumb|400px|'''Graph of quartic function that is quadratic in <math>x^2</math>.'''</br>Because coefficient <math>d</math> is missing, there is a stationary point where <math>x = 0.</math>]] If coefficients <math>b, d</math> are missing, the quartic function becomes a quadratic in <math>x^2.</math> The curve (red line) in diagram has equation: <math>y = f(x) = \frac{x^4 - 13x^2 + 36}{5}</math> The quartic equation may be solved as: <math>X^2 - 13X + 36 = 0</math> where <math>X = x^2</math> or <math>x = \sqrt{X}.</math> <math>X = 4</math> or <math>X = 9.</math> <math>x = \pm 2</math> or <math>x = \pm 3.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0312quartic03.png|thumb|400px| ]] The quartic function may be expressed as <math>x = ay^4 + by^3 + cy^2 + dy + e.</math> Unless otherwise noted, references to "quartic function" on this page refer to function of form <math>y = ax^4 + bx^3 + cx^2 + dx + e.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0318quartic01.png|thumb|400px|'''Graph of quartic function with coefficient a negative.'''</br>There is no absolute minimum.]] Coefficient <math>a</math> may be negative as shown in diagram. As <code>abs</code><math>(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^4.</math> When <code>abs</code><math>(x)</math> is very large, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "quartic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0502quartic01.png|thumb|400px|'''Graph of quartic function with coefficient b missing.''' </br> Sum of roots is <math>0.</math> </br> <math>Y</math>axis compressed for clarity. ]] When sum of roots is <math>0,</math> coefficient <math>b = 0.</math> In the diagram, roots of <math>f(x)</math> are <math>-5, -4, 2, 7.</math> Sum of roots <math>= 0.</math> Therefore coefficient <math>b = 0.</math> {{RoundBoxBottom}} ==Function as product of linear function and cubic== {{RoundBoxTop|theme=3}} [[File:0313quartic01.png|thumb|400px|'''Graphs of quartic function and associated cubic function.''']] When <math>p</math> is a root of the function, the function may be expressed as: <math>(x - p)( Ax^3 + Bx^2 + Cx + D )</math> where <math>A = a;\ B = Ap + b;\ C = Bp + c;\ D = Cp + d.</math> When one real root <math>p</math> is known, the other three roots may be calculated as roots of the cubic function <math>Ax^3 + Bx^2 + Cx + D.</math> In the diagram the quartic function has equation: <math>y = \frac{ x^4 - 23x^3 + 163x^2 - 393x + 252 }{ 48 }.</math> It is known that <math>3</math> is a root of this function. The associated cubic has equation: <math>y = \frac { x^3 - 20x^2 + 103x - 84} { 48 }</math> The 2 curves coincide at points <math>(1, 0),\ (7, 0),\ (12, 0),</math> the three points that are roots of both functions. <math></math> {{RoundBoxBottom}} ==Function defined by 5 points== {{RoundBoxTop|theme=2}} [[File:0313quartic02.png|thumb|400px|'''Figure 1. Quartic function defined by 5 points''' </br>Any 5 points on the curve may be used to define the function.]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 5 unique points on the curve. For example, let us choose the five points: <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code points = (-5,0), (-2,0), (1,0), (3,-6), (6,2) L11 = [] for point in points : x,y = point L11 += [[x*x*x*x, x*x*x, x*x, x, 1, -y]] print (L11) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ 625.0, -125.0, 25.0, -5.0, 1.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 0.0], # [ 1.0, 1.0, 1.0, 1.0, 1.0, 0.0], # matrix supplied to function solveMbyN() below. [ 81.0, 27.0, 9.0, 3.0, 1.0, 6.0], # 5 rows by 6 columns. [1296.0, 216.0, 36.0, 6.0, 1.0, -2.0]] # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L11) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.02651515151515152, 0.004545454545454542, -0.847727272727273, -0.728787878787879, 1.5454545454545459) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by the 5 points <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> is <math>y = \frac{0.875 x^4 + 0.15 x^3 - 27.975 x^2 - 24.05 x + 51}{33}.</math> {{RoundBoxBottom}} ==Function defined by 3 points and 2 slopes== {{RoundBoxTop|theme=2}} [[File:0314quartic03.png|thumb|400px|'''Figure 1. Quartic function defined by 3 points and 2 slopes.''' </br>Any 3 points on the curve and the slopes at any 2 of these points may be used to define the function.</br></br>Slope at <math>(-2, -2)\ =</math> slope at <math>(6, -4)\ =\ 0.</math>]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 3 unique points on the curve and the slopes at any 2 of these points. For example, let us choose the three points: <math>(-2, -2), (6, -4), (4, 1)</math> At point <math>(-2, -2)</math> slope is <math>0.</math> At point <math>(6, -4)</math> slope is <math>0.</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> Rearrange the standard cubic function of slope to prepare for the calculation of <math>a,b,c,d,e:</math> <math> 4 x^3 a + 3 x^2 b + 2 x c + 1 d + 0 e - s = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code def makeEntry(input) : x,y,s = ( tuple(input) + (None,) )[:3] L1 = [] if s != None : L2 = [ float(v) for v in [4*x*x*x, 3*x*x, 2*x, 1, 0, -s] ] L1 += [ L2 ] L2 = [ float(v) for v in [x*x*x*x, x*x*x, x*x, x, 1, -y]] L1 += [ L2 ] return L1 t1 = ( (-2,-2, 0), # point (-2, -2) with slope 0. (6,-4, 0), # point (6, -4) with slope 0. (4,1), # point (4, 1) ) L1 = [] for v in t1 : L1 += makeEntry ( v ) print (L1) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ -32.0, 12.0, -4.0, 1.0, 0.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 2.0], # [ 864.0, 108.0, 12.0, 1.0, 0.0, 0.0], # matrix supplied to function solveMbyN() below. [1296.0, 216.0, 36.0, 6.0, 1.0, 4.0], # 5 rows by 6 columns. [ 256.0, 64.0, 16.0, 4.0, 1.0, -1.0]]. # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L1) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.03255208333333339, -0.2526041666666665, -0.3072916666666667, 2.84375, 2.375) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by three points and two slopes is: <math>y = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> ===Associated cubic functions=== {{RoundBoxTop|theme=3}} ====When p == -2==== {{RoundBoxTop|theme=1}} [[File:0314_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = -2.''' </br>In this case roots of associated cubic include x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == -2,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 15.25 x^2 + 15.75 x + 105 } {48}.</math> Three blue vertical lines show 3 values of <math>x</math> where <math>g(x) = 0</math> and <math>f(x) = f(-2)</math> In this case roots of <math>g(x)</math> include <math>x = p.</math> {{RoundBoxBottom}} ====When p == 5==== {{RoundBoxTop|theme=1}} [[File:0314_2curves02.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 5.''' </br>In this case the one root of associated cubic excludes x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 5,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 4.3125 x^2 - 36.3125 x - 45.0625 } {48}.</math> Two blue vertical lines show 2 values of <math>x</math> where <math>f(x) = f(5)</math> In this case the one root of <math>g(x)</math> excludes <math>x = p.</math> {{RoundBoxBottom}} ====When p == 6==== {{RoundBoxTop|theme=1}} [[File:0315_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 6.''' </br>In this case the one root of associated cubic includes x = p.]] Quartic function is: <math>y = f(x) </math><math>= \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 6,</math> associated cubic function is: <math>y = g(x) </math><math>= \frac{ 1.5625 x^3 - 2.75 x^2 - 31.25 x - 51 } {48}.</math> One blue vertical line shows 1 value of <math>x</math> where <math>f(x) = f(6)</math> In this case the one root of <math>g(x)</math> includes <math>x = p.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} ==Examples== {{RoundBoxTop|theme=2}} ===Quartic with 2 stationary points=== {{RoundBoxTop|theme=1}} [[File:0315_2statPoints02.png|thumb|400px|'''Graph of quartic function with 2 stationary points.''']] In the diagram the red line represents quartic function <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 2 roots of <math>g(x),\ -1</math> and <math>\frac{-1}{4}</math> show that <math>f(x)</math> has stationary points at <math>x = -1</math> and <math>x = -0.25.</math> {{RoundBoxBottom}} ===Quartic with 1 stationary point=== {{RoundBoxTop|theme=1}} [[File:0315_1statPoint.png|thumb|400px|'''Graph of quartic function with 1 stationary point.''']] In the diagram the red line represents quartic function <math>y = f(x) = \frac{x^4 + 3x^3 + 3x^2 + 3x}{2}</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 1 root of <math>g(x),\ -1.607</math> (approx.), shows that <math>f(x)</math> has 1 stationary point where <math>g(x) = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =First and second derivatives= {{RoundBoxTop|theme=1}} ==Points of inflection== {{RoundBoxTop|theme=2}} [[File:0317_3curves01.png|thumb|400px|'''Graphs of quartic function and first two derivatives.'''</br>Dotted portion of black line shows where f(x) is always concave down.</br>Dotted portion of red line shows where g(x) is decreasing.</br>Dotted portion of blue line shows where h(x) is negative.</br>When h(x) is negative, f(x) is concave down.</br>When h(x) is positive, f(x) is concave up.]] In the diagram the black line has equation: <math>y = f(x) = \frac{x^4 - 16x^3 + 42x^2 + 12x + 4}{144}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{x^3 - 12x^2 + 21x + 3}{36}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{x^2 - 8x + 7}{12}.</math> When <math>x < x_1:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. When <math>x == x_1:</math> * <math>y'</math> is at a local maximum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between up and down. When <math>x_1 < x < x_2:</math> * <math>y'</math> is decreasing. * <math>y''</math> is negative. * <math>f(x)</math> is always concave down. When <math>x == x_2:</math> * <math>y'</math> is at a local minimum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between down and up. When <math>x_2 < x:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. {{RoundBoxTop|theme=1}} The roots of <math>h(x): x_1 = 1, x_2 = 7.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At point <math>p_1</math> concavity of <math>f(x)</math> changes from up to down. At point <math>p_2</math> concavity of <math>f(x)</math> changes from down to up. The points <math>p_1, p_2</math> (the <math>X</math> coordinates of which are roots of <math>h(x)</math>) are the '''points of inflection''' of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Maxima and minima== {{RoundBoxTop|theme=2}} [[File:0317_3curves03.png|thumb|400px|'''Graphs of quartic function and first two derivatives showing maximum and minima.'''</br> Point <math>p_{2}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{2}</math> is concave down. Point <math>p_{2}</math> is local maximum. </br> Point <math>p_{1}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{1}</math> is concave up. Point <math>p_{1}</math> is local minimum. </br> Similarly, point <math>p_{3}</math> is local minimum. ]] In the diagram the black line has equation: <math>y = f(x) = \frac{1.5625x^4 - 12.125x^3 - 14.75x^2 + 136.5x + 114}{48}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{6.25x^3 - 36.375x^2 - 29.5x + 136.5}{48}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{18.75x^2 - 72.75x - 29.5}{48}.</math> Roots of <math>g(x):\ x_1 = -2;\ x_2 = 1.82;\ x_3 = 6.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> At <math>x_1\ h(x_1)</math> is positive. Point <math>p_1</math> is a stationary point and <math>f(x)</math> at <math>p_1</math> is concave up. Point <math>p_1</math> is a local minimum. Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At <math>x_2\ h(x_2)</math> is negative. Point <math>p_2</math> is a stationary point and <math>f(x)</math> at <math>p_2</math> is concave down. Point <math>p_2</math> is a local maximum. Let point <math>p_3</math> on <math>f(x)</math> have coordinates <math>(x_3, f(x_3)).</math> At <math>x_3\ h(x_3)</math> is positive. Point <math>p_3</math> is a stationary point and <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is a local minimum. {{RoundBoxBottom}} ==Quartic with 2 stationary points== {{RoundBoxTop|theme=3}} [[File:0318_3curves01.png|thumb|400px|'''Graph of quartic function with 2 stationary points and first 2 derivatives.''' </br> Black line: <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> </br> Red line: <math>y' = g(x) = 4(4x^3 + 9x^2 + 6x + 1).</math> </br> <math>y'' = h(x) = 4(12x^2 + 18x + 6)</math> </br> Blue line: <math>y'' = 4(2x^2 + 3x + 1) = \frac{h(x)}{6}.</math> </br> Dotted portion of black line shows where <math>f(x)</math> is concave down. ]] In the diagram, point <math>p_1</math> on <math>f(x)</math> has coordinates <math>(x_1, f(x_1)).</math> Similarly, points <math>p_2, p_3</math> have coordinates <math>(x_2, f(x_2)),\ (x_3, f(x_3)).</math> <math>y'</math> has roots: <math>x_1 = -1;\ x_3 = -0.25.</math> Points <math>p_1, p_3</math> are stationary points. <math>y''</math> has roots: <math>x_1 = -1;\ x_2 = -0.5.</math> Points <math>p_1, p_2</math> are points of inflection. At point <math>p_3\ y''</math> is positive. <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is local minimum. {{RoundBoxTop|theme=2}} Summary: * Point <math>p_1</math> is both stationary point and point of inflection. * Point <math>p_2</math> is point of inflection. * Point <math>p_3</math> is both stationary point and local minimum. {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =The simplest quartic function= {{RoundBoxTop|theme=2}} [[File:0320quartic01.png|thumb|400px|'''Graph of simplest quartic function.''' </br> Point <math>p_1\ (-1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_2\ (1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_0\ (0, -1.1^4)</math> is <math>Y</math> intercept of <math>f(x).</math> ]] The simplest quartic function has coefficients <math>b = c = d = 0.</math> Red line in diagram has equation: <math>y = f(x) = x^4 - 1.1^4</math> First derivative (not shown) of <math>f(x):\ y' = g(x) = 4x^3.</math> When <math>x == 0,\ g(x) = 0.</math> There is a stationary point on <math>f(x)</math> when <math>x == 0,</math> point <math>p_0.</math> Second derivative (not shown) of <math>f(x):\ y'' = h(x) = 12x^2.</math> When <math>x == 0,\ h(x) = 0.</math> There is a point of inflection on <math>f(x)</math> when <math>x == 0.</math> For every non-zero value of <math>x,\ h(x)</math> is positive. To left and right of point <math>p_0,\ f(x)</math> is always concave up. Point <math>p_0</math> is both local minimum and absolute minimum. * Point <math>p_0</math> is stationary point and point of inflection and absolute minimum. Curve <math>f(x)</math> is useful for finding the fourth root of a real number. Solve: <math>x = N^{\frac{1}{4}}.</math> <math>x^4 = N.</math> <math>x^4 - N = 0.</math> This is equivalent to finding a root of function <math>y = j(x) = x^4 - N.</math> If you use Newton's method to find a root of <math>j(x),</math> this would be more efficient than solving <math>x = \sqrt{\sqrt{N}}.</math> {{RoundBoxBottom}} =Roots of equal absolute value= {{RoundBoxTop|theme=1}} The standard quartic function: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> For <math>x</math> in <math>(1)</math> substitute <math>(p+q).</math> Call this <math>(2).</math> For <math>x</math> in <math>(1)</math> substitute <math>(p-q).</math> Call this <math>(3).</math> Combine <math>(2)</math> and <math>(3)</math> to eliminate <math>q</math> and produce an equation in <math>p:</math> <math>(- 64aaa)pppppp+</math> <math>(- 96aab)ppppp+</math> <math>(- 32aac - 48abb)pppp+</math> <math>(- 32abc - 8bbb)ppp+</math> <math>(+ 16aae - 4abd - 4acc - 8bbc)pp+</math> <math>(+ 8abe - 2bbd - 2bcc)p+</math> <math>(+ add + bbe - bcd)\ =\ 0\ \dots\ (4).</math> We are interested in coefficient <math>0</math> of <math>(4):\ c_0 = add + bbe - bcd.</math> If <math>c_0 == 0,\ p=0</math> is a solution and function <math>(1)</math> has 2 roots of form <math>0 \pm q</math> where <math>q = \sqrt{\frac{-d}{b}}.</math> An example: {{RoundBoxTop|theme=3}} [[File:0320quartic02.png|thumb|400px|'''Graph of quartic function with 2 roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 12x^3 + 31x^2 + 48x -140}{45}.</math> <math>a,b,c,d,e = 1,-12,31,48,-140</math> <math>c_0 = add + bbe - bcd </math><math>= 1(48)(48) + (-12)(-12)(-140) - (-12)(31)(48) = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-48}{-12}} = \sqrt{4} = \pm 2.</math> The 2 roots of equal absolute value are: <math>2, -2.</math> {{RoundBoxBottom}} The method works with complex roots of equal absolute value: {{RoundBoxTop|theme=2}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with 2 complex roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}.</math> <math>a,b,c,d,e = 1,-3,-1,-27,-90</math> <math>c_0 = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-(-27)}{-3}} = \sqrt{-9} = \pm 3i.</math> The 2 roots of equal absolute value are: <math>3i, -3i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Equal roots= {{RoundBoxTop|theme=2}} Equal roots occur when the function and the slope of the function both equal zero. <math>ax^4 + bx^3 + cx^2 + dx + e = 0\ \dots\ (1)</math> <math>4ax^3 + 3bx^2 + 2cx + d = 0\ \dots\ (2)</math> Begin the process of reducing <math>(1),\ (2)</math> to linear functions. Combine <math>(1),\ (2)</math> to produce 2 cubic functions: <math>Fx^3 + Gx^2 + Hx + J\ \dots\ (1a)</math> where: <math>F = ad;\ G = bd - 4ae;\ H = cd - 3be;\ J = dd - 2ce.</math> <math>fx^3 + gx^2 + hx + j\ \dots\ (2a)</math> where: <math>f = 4a;\ g = 3b;\ h = 2c;\ j = d.</math> Combine <math>(1a),\ (2a)</math> to produce 2 quadratic functions: <math>Kx^2 + Lx + M\ \dots\ (1b)</math> where: <math>K = Gf -Fg;\ L = Hf -Fh;\ M = Jf - Fj.</math> <math>kx^2 + lx + m\ \dots\ (2b)</math> where: <math>k = Fj - Jf;\ l = Gj - Jg;\ m = Hj - Jh.</math> Combine <math>(1b),\ (2b)</math> to produce 2 linear functions: <math>Rx + S\ \dots\ (1c)</math> where: <math>R = Lk - Kl;\ S = Mk - Km.</math> <math>rx + s\ \dots\ (2c)</math> where: <math>r = Km-Mk;\ s = Lm - Ml.</math> From <math>(1c):\ x_1 = \frac{-S}{R}</math> From <math>(2c):\ x_2 = \frac{-s}{r}</math> If <math>x_1 == x_2:</math> <math>\frac{-S}{R} = \frac{-s}{r}</math> <math>Rs = rS</math> <math>Rs - Sr = 0.</math> The value <math>Rs - Sr</math> is in fact: <syntaxhighlight> + 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee - 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd + 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee + 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde + 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee - 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde - 3aabbccdddddd </syntaxhighlight> which, by removing values <math>aa, d</math><math>d</math> (common to all values), may be reduced to: <syntaxhighlight> status = ( + 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee - 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd + 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee + 384abccddde - 54abcddddd + 128acccccee - 80accccdde + 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee - 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde - 3bbccdddd ) </syntaxhighlight> If <math>status == 0,</math> there are at least 2 equal roots which may be calculated as shown below. {{RoundBoxTop|theme=2}} If coefficient <math>d</math> is non-zero, it is not necessary to calculate <math>status.</math> If coefficient <math>d == 0,</math> verify that <math>status = 0</math> before proceeding. {{RoundBoxBottom}} ==No equal roots== {{RoundBoxTop|theme=1}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with no equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}</math> <math>a,b,c,d,e = 1, -3, -1, -27, -90</math> <math>R, S = 15269148, -35977608</math> <math>x_1 = \frac{-S}{R} = \frac{35977608}{15269148} = 2.3562289133617\dots</math> <math>r, s = 35977608, -60634332</math> <math>x_2 = \frac{-s}{r} = \frac{60634332}{35977608} = 1.685335278543253\dots</math> <math>x_1 != x_2.</math> There are no equal roots. {{RoundBoxBottom}} ==Exactly 2 equal roots== {{RoundBoxTop|theme=1}} [[File:0329quartic04.png|thumb|400px|'''Graph of quartic function with exactly 2 equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 + 6x^3 - 48x^2 - 182x + 735}{100}</math> <math>a,b,c,d,e = 1, 6, -48, -182, 735</math> <math>R, S = -1027353600, -7191475200</math> <math>x_1 = \frac{-S}{R} = \frac{7191475200}{-1027353600} = -7</math> <math>r, s = 7191475200, 50340326400</math> <math>x_2 = \frac{-s}{r} = \frac{-50340326400}{7191475200} = -7</math> <math>x_1 = x_2 = -7.</math> There are 2 equal roots at <math>x = -7.</math> {{RoundBoxTop|theme=2}} The following 3 graphs show the steps that lead to calculation of equal roots at point <math>(-7, 0).</math> <gallery> File:0914quartic00.png|<small>Figure 1. graph of <math>f(x)</math> and 2 associated cubic functions.</small> File:0914quartic01.png|<small>Figure 2. graph of <math>f(x)</math> and 2 associated quadratic functions.</small> File:0914quartic02.png|<small>Figure 3. graph of <math>f(x)</math> and 2 associated linear functions.</small> </gallery> In all graphs, all curves have a common root at point <math>(-7, 0).</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=4}} See [[Quartic_function#Function_as_product_of_linear_function_and_cubic | "Function as product of linear function and cubic"]] above. To calculate all roots: <syntaxhighlight lang=python> # python code. a,b,c,d,e = 1,6,-48,-182,735 # The associated cubic: p = -7 A = a B = A*p + b C = B*p + c D = C*P + d # The associated quadratic: a1 = A b1 = a1*p + B c1 = b1*p + C a1,b1,c1 </syntaxhighlight> <syntaxhighlight> (1, -8, 15) </syntaxhighlight> Roots of quadratic function <math>g(x) = x^2 - 8x + 15</math> are <math>3, 5.</math> All roots of <math>f(x)</math> are <math>-7, -7, 3, 5.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Exactly 3 equal roots== {{RoundBoxTop|theme=1}} [[File:0320_2curves01.png|thumb|400px|'''Graph of quartic function with exactly 3 equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 2x^3 - 36x^2 + 162x - 189}{100}</math> <math>a,b,c,d,e = 1, -2, -36, 162, -189</math> <math>R, S = 0, 0\ \dots\dots\ (1c)</math> <math>r, s = 0, 0\ \dots\dots\ (2c)</math> In this case the calculation of <math>x_1, x_2</math> is not appropriate because there are more than 2 equal roots. Try equations <math>(1b), (2b).</math> Both of these are equivalent to: <math>y = g(x) = x^2 - 6x + 9,</math> blue line in diagram. Discriminant of <math>g(x) = (-6)^2 - 4(1)(9) = 0.\ g(x)</math> has two equal roots at <math>x = \frac{-(-6)}{2(1)} = 3.</math> Therefore <math>f(x)</math> has 3 equal roots at <math>x = 3.</math> {{RoundBoxBottom}} ==Four equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves01.png|thumb|400px|'''Graph of quartic function with 4 equal roots and corresponding cubic.''' </br> <math>g(x) = \frac{f'(x)}{4}.</math> ]] Red line in diagram is of function: <math>y = f(x) = x^4 - 20x^3 + 150x^2 - 500x + 625.</math> <math>a,b,c,d,e = 1, -20, 150, -500, 625</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> <math>K, L, M = 0, 0, 0</math> <math>k, l, m = 0, 0, 0</math> In this case <math>(1b), (2b), (1c), (2c)</math> are all null. This is the only case in which <math>(1b), (2b)</math> are null. <math>(1a), (2a)</math> are both equivalent to: <math>y = g(x) = x^3 - 15x^2 + 75x - 125,</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 4 equal roots at <math>x = 5.</math> {{RoundBoxBottom}} ==Two pairs of equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 6x^3 - 11x^2 + 60x + 100}{20}.</math> <math>a,b,c,d,e = 1, -6, -11, 60, 100</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> In this case <math>(1c), (2c)</math> are both null. <math>(1b), (2b)</math> are both equivalent to: <math>y = g(x) = \frac{x^2 - 3x - 10}{20},</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = -2</math> and one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 2 equal roots at <math>x = -2</math> and 2 equal roots at <math>x = 5.</math> {{RoundBoxTop|theme=2}} [[File:0408_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal, complex roots and corresponding quadratic.''' </br> <math>g(x) = (x-(3-2i))(x-(3+2i)) = x^2 - 6x + 13.</math> </br> <math>f(x) = (g(x))^2.</math> ]] This method is valid for complex roots. For example: <math>y = f(x) = x^4 - 12x^3 + 62x^2 - 156x + 169.</math> <math>a,b,c,d,e = 1,-12,62,-156,169.</math> In this case <math>(1c),\ (2c)</math> are both null. <math>(1b),\ (2b)</math> are both equivalent to: <math>y = g(x) = x^2 - 6x + 13,</math> blue line in diagram. Roots of <math>g(x)</math> are <math>3 \pm 2i.</math> <math>f(x)</math> has 2 roots equal to <math>3 + 2i</math> and 2 roots equal to <math>3 - 2i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Summmary== {{RoundBoxTop|theme=1}} {| class="wikitable" |- ! || No equal roots !! 2 equal roots !! 3 equal roots !! 4 equal roots !! 2 pairs of equal roots |- | Cubic: 1(a), 2(a) | different | different | different | same | different |- | Quadratic: 1(b), 2(b) | different | different | same, 1root | null | same, 2roots |- | Linear: 1(c), 2(c) | different | same | null | null | null |} {{RoundBoxBottom}} ==Caution== {{RoundBoxTop|theme=4}} [[File:0915quartic02.png|thumb|400px| Calculation of false equal roots.</br> In this example, method calculates 2 legitimate equal roots at <math>(4, 0)</math> and 2 false equal roots at <math>(7.543296089385474,0).</math> <math></math> ]] Black line in diagram has equation: <math>y = f(x) = 0.012684240362811794x^4</math> <math> -\ 0.19522392290249435x^3 </math> <math> +\ 0.7654478458049887x^2 + 0x - 3. </math> <math>f(x)</math> is a quartic function with exactly 2 equal roots and coefficient <math>d</math> missing. Calculation of equal roots of <math>f(x)</math> gives linear functions <math>(1c), (2c)</math> null and quadratic functions <math>(1b), (2b)</math> with equal roots of <math> (4,0), (7.543296089385474,0). </math> Usually, this indicates that <math>f(x)</math> should have 2 equal roots at <math>(4,0)</math> and 2 equal roots at <math>(7.543296089385474,0).</math> It is obvious that <math>7.543296089385474</math> is not a root of <math>f(x).</math> When <math>x = 7.543296089385474,</math> slope of derivative <math>g(x) = 0.</math> Value of <math>f(x)\ != 0.</math> This example indicates that it would be wise to verify that calculated equal roots are in fact valid roots of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Depressed quartic= {{RoundBoxTop|theme=1}} A depressed quartic is any quartic function with any one or more of coefficients <math>b,c,d</math> missing. Within this section a depressed quartic has coefficient <math>b</math> missing. To produce the depressed quartic: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> <math>y = \frac{(4^4 a^3)(ax^4 + bx^3 + cx^2 + dx + e)}{4^4 a^3}\ \dots\ (2)</math> Let <math>x = \frac{-b + t}{4a}.</math> Substitute in <math>(2),</math> expand and simplify: <math>y = \frac{t^4 + At^2 + Bt + C}{4^4a^3}\ \dots\ (3)</math> where: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> When equated to <math>0,\ (3)</math> becomes the depressed equation: <math>t^4 + At^2 + Bt + C = 0\ \dots\ (4).</math> Be prepared for the possibility that any 1 or more of <math>A, B, C</math> may be zero. ==Coefficient B missing== {{RoundBoxTop|theme=2}} [[File:0322_3curves01.png|thumb|400px|'''Graph of quartic function that resembles a quadratic.''' ]] If coefficient <math>B == 0,\ (4)</math> becomes a quadratic in <math>t^2:</math> <math>t^4 + At^2 + C = 0.</math> <math>(1)</math> has the appearance of a quadratic. The black line: <math>y = f(x) = \frac{x^4 - 4x^3 + 9x^2 - 10x + 5}{10}</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>= 8( 8(-10) - 4(-4)(9) + -64 )</math> <math>= 8(-80 + 144 - 64)</math> <math>= 8(0) = 0.</math> The red line: <math>y' = g(x) = \frac{4x^3 - 12x^2 + 18x - 10}{10}</math> <math>y' = g(x) = 0</math> where <math>x = 1.</math> The grey line: <math>y'' = h(x) = \frac{12x^2 - 24x + 18}{10}</math> * Absolute minima of <math>f(x)</math> and of <math>h(x)</math> and point of inflection of <math>g(x)</math> occur where <math>x = \frac{-b}{4} = 1.</math> * <math>y''</math> is always positive. <math>f(x)</math> is always concave up. * <math>f(1 + p) = f(1-p).</math> {{RoundBoxTop|theme=3}} If <math>(1)</math> contains 2 pairs of equal roots, coefficient <math>B = 0.</math> The converse is not necessarily true. If <math>(1)</math> contains 4 equal roots, coefficients <math>A = B = C = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Coefficient C missing== {{RoundBoxTop|theme=2}} [[File:0501quartic01.png|thumb|400px|'''Graph of quartic function with coefficient C of depressed function missing.''' </br> <math>Y</math> axis compressed for clarity. ]] If coefficient <math>C == 0,\ (4)</math> becomes: <math>t^4 + At^2 + Bt = t(t^3 + At + B) = 0</math> in which case <math>t = 0</math> is a solution and <math>x = \frac{-b}{4a}</math> is a root of <math>(1).</math> Curve (red line) in example has equation: <math>y = f(x) = 8x^4 + 16x^3 + 24x^2 + 89x + 40.</math> Coefficients of depressed function are: <syntaxhighlight lang=python> # python code a,b,c,d,e = 8, 16, 24, 89, 40 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b A,B,C </syntaxhighlight> <syntaxhighlight> (1536, 299008, 0) </syntaxhighlight> Coefficient <math>C</math> of depressed function is missing. <math>t = 0</math> is a solution. Using <math>x = \frac{-b + t}{4a},</math> one root of <math>f(x) = \frac{-16 + 0}{4(8)} = -0.5.</math> {{RoundBoxBottom}} ==Resolvent cubic== {{RoundBoxTop|theme=1}} This section introduces a special cubic function called "resolvent" because it helps to resolve a requirement, the calculation of the roots of the quartic. The depressed quartic: <math>t^4 + At^2 + Bt + C\ \dots\ (1)</math> For <math>t</math> substitute <math>(u+v)\ \dots\ (2)</math> For <math>t</math> substitute <math>(u-v)\ \dots\ (3)</math> <math>(2)+(3):\ 2Auu + 2Avv + 2Bu + 2C + 2uuuu + 12uuvv + 2vvvv\ \dots\ (4)</math> Simplify <math>(4):\ Auu + AV + Bu + C + uuuu + 6uuV + VV\ \dots\ (4a)</math> <math>(2)-(3):\ 4Auv + 2Bv + 8uuuv + 8uvvv\ \dots\ (5)</math> Simplify <math>(5):\ 2Au + B + 4uuu + 4uV\ \dots\ (5a)</math> From <math>(5a):\ 4uV = -(2Au + B + 4uuu)\ \dots\ (5b)</math> <math>(4a)*4u4u:\ 4u4uAuu + A4u(4uV) + 4u4uBu + 4u4uC + 4u4uuuuu + 6uu4u(4uV) + (4uV)(4uV)\ \dots\ (6)</math> In <math>(6)</math> replace <math>4uV</math> with <math>(-(2Au + B + 4uuu)),</math> expand, simplify, gather like terms and result is: <math>Pu^6 + Qu^4 + Ru^2 + S</math> or {{RoundBoxTop|theme=2}} <math>PU^3 + QU^2 + RU + S\ \dots\ (7)</math> where: <math>U = u^2</math> <math>P = 64</math> <math>Q = 32A</math> <math>R = 4A^2 - 16C</math> <math>S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} From <math>(5b):\ V = v^2 = \frac{-(2Au + B + 4uuu)}{4u} = -(\frac{A}{2} + U) - \frac{B}{4u}\ \dots\ (8)</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Some simple changes reduce the number of calculations and also the sizes of coefficients <math>P, Q, R, S.</math> <math>A2 = 8ac - 3b^2</math> where <math>A2 = \frac{A}{2}</math> <math>B4 = 16a^2 d - 8abc + 2b^3</math> where <math>B4 = \frac{B}{4}</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> Then: <math>P = 64</math> <math>Q = 32(A2)(2) = 64A2</math> <math>R = 4(A2)(2)(A2)(2) - 16C = 16A2^2 - 16C</math> <math>S = -(B4)(4)(B4)(4) = -16 B4^2</math> Divide all 4 coefficients by <math>16:</math> <math>P = 4</math> <math>Q = 4A2</math> <math>R = A2^2 - C</math> <math>S = -B4^2</math> <math>V = v^2 = -(A2 + U) - \frac{B4}{u}.</math> A close examination of coefficients <math>R, S</math> shows that both coefficients are always exactly divisible by <math>4.</math> Therefore, all four coefficients may be defined as follows: <math>P = 1</math> <math>Q = A2</math> <math>R = \frac{A2^2 - C}{4}</math> <math>S = \frac{-B4^2}{4}</math> Actual calculations of values <math>R, S</math> are: <syntaxhighlight lang=python> # python code. R = ( (16)*(a)*(a)*(b)*(d) + (16)*(a)*(a)*(c)*(c) + (3)*(b)*(b)*(b)*(b) + (-16)*(a)*(b)*(b)*(c) + (-64)*(a)*(a)*(a)*(e) ) S = ( (64)*(a)*(a)*(a)*(b)*(c)*(d) + (8)*(a)*(b)*(b)*(b)*(b)*(c) + (-16)*(a)*(a)*(b)*(b)*(b)*(d) + (-16)*(a)*(a)*(b)*(b)*(c)*(c) + (-1)*(b)*(b)*(b)*(b)*(b)*(b) + (-64)*(a)*(a)*(a)*(a)*(d)*(d) ) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =Solving quartic equation= This section presents 4 examples that show how to use the depressed quartic and the resolvent cubic to solve the quartic equation. ==Four real roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves01.png|thumb|400px|'''Graphs of quartic function with 4 real roots and associated resolvent cubic.''' </br>Resolvent cubic contains 3 real, positive roots. ]] Calculate roots of: <math>y = f(x) = x^4 - x^3 - 19x^2 - 11x + 30</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,-1,-19,-11,30 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -310 -1320 6669 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 -9920 277696 -1742400 </syntaxhighlight> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = 64x^3 - 9920x^2 + 277696x - 1742400.</math> There are 3 real, positive roots: <math>9, 25, 121.</math> Using 3 roots of <math>g(x),</math> calculate 4 roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code for U in (9, 25, 121) : print ('\nU =', U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** .5 for t in (u+v, u-v) : x = (-b+t) / (4*a) y = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x:',x, '; y:',y) </syntaxhighlight> <syntaxhighlight> U = 9 x: 5.0 ; y: 0.0 x: -3.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 U = 25 x: 5.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -3.0 ; y: 0.0 U = 121 x: 5.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: -3.0 ; y: 0.0 </syntaxhighlight> Roots of <math>f(x)</math> are: <math>5, 1, -2, -3.</math> {{RoundBoxBottom}} All 3 values of <math>U</math> produce the same results, but not in same sequence. It is not necessary to calculate all 3 roots of resolvent cubic. Any one non-zero root is sufficient to do the job. ==Two real and two complex roots== ===Example 1=== {{RoundBoxTop|theme=4}} [[File:0323_2curves02.png|thumb|400px|'''Graphs of quartic function with 2 real and 2 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 + 2x^3 + 18x^2 - 70x - 87</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,2,18,-70,-87 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> 264 -5568 -12208 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 8448 474112 -31002624 </syntaxhighlight> {{RoundBoxBottom}} Calculate one real root of cubic function: <math>y = g(x) = 64x^3 + 8448x^2 + 474112x -31002624.</math> <math>36</math> is one real root. Choose <math>U = 36.</math> Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 36 u1 = U**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) if V >= 0 : v = V**.5 else : v = 1j * (-V)**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =',x, '; sum =',sum ) </syntaxhighlight> <syntaxhighlight> x = 3.0 ; sum = 0.0 x = -1.0 ; sum = 0.0 x = (-2+5j) ; sum = 0j x = (-2-5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=4}} [[File:0321_2curves00.png|thumb|400px| Coefficient <math>d</math> of <math>g(x) = 0.</math> <math></math> ]] Calculate roots of: <math>y = f(x)</math> <math>= 3x^4 - 6x^3 - 41x^2 +44x - 189</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 3, -6, -41, 44, -189 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -2184, 0, -1229040 </syntaxhighlight> Notice that coefficient <math>B = 0.</math> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 1, -1092, 605376, 0 </syntaxhighlight> Notice that coefficient <math>S = 0.</math> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = x^3 - 1092x^2 + 605376x + 0.</math> Roots are <math>0, 546 \pm 554.3103823671355j.</math> Value <math>0</math> cannot be used because it will cause error <code>Divide by zero</code> at statement <code>V = -(A/2 + U) - B/(4*u)</code>. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 546+554.3103823671355j print ('\nU =',U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** 0.5 s1 = '\nu,v' print (s1,eval(s1)) for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> U = (546+554.3103823671355j) u,v ((25.729935131257832+10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (4.788322521876306 + 0j) ; sum = (1.9895196601282805e-13 + 0j) x = (0.5 + 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) u,v ((-25.729935131257832-10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (0.5 - 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) x = (-3.7883225218763052 + 0j) ; sum = (1.7053025658242404e-13 + 0j) </syntaxhighlight> Values of <math>x</math> are: <math>-3.7883225218763052, 4.788322521876306, 0.5 \pm 1.795283650280614j</math> <math></math><math></math> {{RoundBoxBottom}} ====Depressed quartic as quadratic==== In this example coefficient <math>B</math> of depressed quartic <math>= 0.</math> Therefore, resolvent cubic can be ignored and depressed quartic processed as quadratic in <math>T = t^2.</math> <math>t^4 - 2184t^2 + (0)t - 1229040</math> <math>T^2 - 2184T - 1229040</math> where <math>T = t^2.</math> Solutions of this quadratic are: <math>T_1, T_2 = 2648.1182474349434, -464.11824743494344</math> <syntaxhighlight lang=python> T1,T2 = 2648.1182474349434, -464.11824743494344 t1 = T1 ** 0.5; t2 = ((-T2) ** 0.5) * 1j for t in (t1,-t1,t2,-t2) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> x = 4.788322521876305 ; sum = -1.7053025658242404e-13 x = -3.788322521876305 ; sum = -1.7053025658242404e-13 x = (0.5 + 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) x = (0.5 - 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) </syntaxhighlight> or <syntaxhighlight> x = 0.5 ± 4.288322521876305, 0.5 ± 1.7952836502806138j </syntaxhighlight> With precision of 15, values of <math>x</math> are same as those shown above. When roots of quartic function are of form <code>p ± q, p ± r,</code> coefficient <math>B</math> of depressed function <math>= 0.</math> ==Four complex roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves03.png|thumb|400px|'''Graphs of quartic function with 4 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 - 20x^3 + 408x^2 + 2296x + 18020</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 4128 344064 9683200 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 64 132096 -86769664 -118380036096 </syntaxhighlight> {{RoundBoxBottom}} Calculate one root of cubic function: <math>y = g(x) = 64x^3 + 132096x^2 - 86769664x - 118380036096.</math> There are 3 real roots: <math>-2304, -784, 1024.</math> Choose <math>U = -784.</math> Negative <math>U</math> is chosen here to show that any 1 of the roots produces the correct result. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = -784 u1 = 1j * (-U)**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) v = V**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> # python expresses complex numbers with 'j'. x = (13+19j) ; sum = 0j x = (-3-5j) ; sum = 0j x = (13-19j) ; sum = 0j x = (-3+5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} =Quartic formula= The substitutions made above can be used to produce a formula for the solution of the quartic equation. See main articles [https://en.wikipedia.org/wiki/Quartic_equation#The_general_case "The general case" ] or [https://en.wikipedia.org/wiki/Quartic_function#General_formula_for_roots "General formula for roots."] {{RoundBoxTop|theme=2}} Both links above point to formula for equation <math>x^4 + ax^3 + bx^2 + cx + d = 0.</math> {{RoundBoxBottom}} Given quartic equation: <math>ax^4 + bx^3 + cx^2 + dx + e = 0,</math> calculate the 4 values of <math>x.</math> <math>x = \frac{-b + t}{4a}</math> where: {{RoundBoxTop|theme=2}} Coefficients of depressed quartic: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of resolvent cubic: <math>a_1 = P = 64</math> <math>b_1 = Q = 32A</math> <math>c_1 = R = 4A^2 - 16C</math> <math>d_1 = S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of depressed cubic: <math>A_1 = 9a_1 c_1 - 3b_1^2</math> <math>B_1 = 27a_1^2d_1 - 9a_1b_1c_1 + 2b_1^3</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of resolvent cubic: <math>C_1 = \frac{-A_1}{3} = b_1^2 - 3a_1 c_1</math> <math>\Delta = B_1^2 - 4C_1^3\ \dots\dots\ \Delta</math> may be negative. <math>\delta = \sqrt{\Delta}</math> <math>W = \frac{-B_1 + \delta}{2}</math> <math>w = \sqrt[3]{W}</math> <math>t_1 = w + \frac{C_1}{w}</math> <math>U = \frac{-b_1 + t_1}{3a_1}</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of quartic: <math>u = \sqrt{U}\ \dots\dots\ u</math> may be positive or negative. <math>V = -(\frac{A}{2} + U) - \frac{B}{4u}</math> <math>v = \sqrt{V}\ \dots\dots\ v</math> may be positive or negative. <math>t = u + v</math> {{RoundBoxBottom}} Formula above produces one value of <math>x.</math> Python code below utilizes <math>\pm \sqrt{U}</math> and <math>\pm \sqrt{V}</math> to produce 4 values of <math>t</math> and then, four values of <math>x.</math> ==An example:== {{RoundBoxTop|theme=4}} [[File:0330quartic01.png|thumb|400px|'''Graph of quartic function with 2 real roots.'''</br><math>Y</math> axis compressed for clarity.]] Calculate roots of <math>f(x) = 4x^4 + 4x^3 - 75x^2 - 776x - 1869.</math> <syntaxhighlight lang=python> # Python code. a,b,c,d,e = 4, 4, -75, -776, -1869 values_of_t = [ t # Coefficients of depressed quartic: for A in (16*a*c - 6*b*b,) for B in (64*a*a*d -32*a*b*c + 8*b*b*b,) for C in (256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b,) # Coefficients of resolvent cubic: for a1 in (64,) for b1 in (32*A,) for c1 in (4*A*A - 16*C,) for d1 in (-B*B,) for U in [ # The resolvent cubic: (-b1+t1)/(3*a1) # Coefficients of depressed resolvent cubic: for A1 in (9*a1*c1 - 3*b1*b1,) for B1 in (27*a1*a1*d1 - 9*a1*b1*c1 + 2*b1*b1*b1,) # One root of resolvent cubic: for C1 in (-A1/3,) for Δ in (B1*B1 - 4*C1*C1*C1,) for δ in (Δ**0.5,) for W in ((-B1 + δ)/2,) for w in (W**(1/3),) for t1 in (w + C1/w,) # See note below. ] # Prepare to calculate 4 values of t. for u1 in (U**.5,) for v1 in ( -(A/2 + U) ,) # Calculate 4 values of t. for u in (u1, -u1,) for V in ( v1 - B/(4*u),) for v in (V**.5,) for t in (u+v, u-v) ] print ('values_of_t =', values_of_t) </syntaxhighlight> {{RoundBoxTop|theme=4}} <syntaxhighlight> values_of_t = [116, -44, (-36+64j), (-36-64j)] </syntaxhighlight> Because <math>f(t)</math> is a depressed quartic function, sum of four <code>values_of_t</code> <math>= 116 - 44 - 36(2) = 0.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> # Python code. # Calculate 4 separate roots. values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] print ('values_of_x =', values_of_x) </syntaxhighlight> <syntaxhighlight lang=python> values_of_x = [7, -3, (-2.5 + 4j), (-2.5 - 4j)] </syntaxhighlight> In python the imaginary part of a complex number is shown with <math>j</math> instead of <math>i.</math> If <math>A == B == C == 0,\ f(x)</math> contains 4 equal roots and <math>x = \frac{-b}{4a}.</math> If <math>f(x)</math> contains 3 or more equal roots, statement <code>for t1 in (w + C1/w,)</code> fails with divisor <math>w = 0.</math> Before using this formula, check for equal roots as in [[Quartic_function#Exactly_3_equal_roots | "Exactly 3 equal roots"]] above. {{RoundBoxTop|theme=1}} Values displayed above have been edited slightly. Actual calculated values were: <syntaxhighlight> values_of_x = [7.000000000000001, -3.0000000000000044, (-2.499999999999998+4.000000000000001j), (-2.4999999999999987-4.000000000000001j)] </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ==In practice== {{RoundBoxTop|theme=5}} The following Python code implements the quartic formula. However, under statement <code>if B4 == 0 :</code> there is code that processes the depressed quartic as a quadratic in <math>T = t^2.</math> This ensures that execution of formula does not fail with error <code>Divide by zero</code> at statement <code>for V in ( v1 - B4/u,).</code> {{RoundBoxTop|theme=4}} <syntaxhighlight lang=python> # python code import cmath cxSqrt = cmath.sqrt # Square root of complex number. def rootsOfQuartic (abcde) : ''' x1,x2,x3,x4 = rootsOfQuartic ((a,b,c,d,e)) Each member of input must be int or float or Decimal object. Int or Decimal object in input is quietly converted to float. Output may be None. ''' def formatResults (x1x2x3x4) : ''' This function improves appearance of results. (8 + 0j) becomes 8.0 ''' values_of_x = list (x1x2x3x4) for p in (0,1,2,3) : v = values_of_x[p] if isinstance (v, complex) and (v.imag == 0) : values_of_x[p] = v.real return values_of_x status = 0 try : a,b,c,d,e = [ float(v) for v in abcde ] except : status = 1 if status : print ('rootsOfQuartic () 1: Error creating coefficients a,b,c,d,e.') return None if a == 0 : print ('rootsOfQuartic () 2: Coefficient a must be non-zero.') return None # Coefficients of depressed quartic, modified. # A = 16*a*c - 6*b*b A2 = 8*a*c - 3*b*b # B = 64*a*a*d - 32*a*b*c + 8*b*b*b B4 = 16*a*a*d - 8*a*b*c + 2*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b if B4 == 0 : # B = 0. # Result returned from this section is type tuple, indicating that coefficient B4 = 0. if A2==C==0 : # 4 equal roots. root = -b/(4*a) return tuple(formatResults((root,root,root,root))) # t**4 + At**2 + (0)t + C # Depressed quartic is quadratic in T: # T**2 + AT + C where T = t**2 # T**2 + 2(A2)T + C where A = 2(A2) # -2(A2) +/- (4(A2)(A2) - 4C)**0.5 # T = -------------------------------- = -A2 +/- ((A2)(A2) - C)**0.5 # 2 disc = A2*A2 - C if disc >= 0 : root = disc ** 0.5 else : root = ((-disc) ** 0.5) * 1j T1 = -A2 - root ; T2 = -A2 + root t1 = cxSqrt(T1) ; t2 = cxSqrt(T2) values_of_t = (t1,-t1,t2,-t2) values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] return tuple(formatResults(values_of_x)) # B4 is non-zero. Therefore, all of (S, U, u) are non-zero. P,Q,R,S = 1, A2, (A2*A2 - C)/4, -B4*B4/4 # str1 = 'P,Q,R,S' ; print (str1, eval(str1)) U = oneRootOfCubic((P,Q,R,S)) # Resolvent cubic. if U > 0 : sqrtU = U ** 0.5 elif U == 0 : # This should not happen. print ('rootsOfQuartic () 3: Internal error.') return None else : sqrtU = ((-U) ** 0.5) * 1j v1 = -(A2+U) values_of_t = [ t for u in (sqrtU, -sqrtU) for V in ( v1 - B4/u, ) for v in ( cxSqrt(V), ) for t in (u+v, u-v) ] values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] # Result returned from this section is type list, indicating that coefficient B4 != 0. return formatResults(values_of_x) </syntaxhighlight> For function <code>oneRootOfCubic()</code> see [[Cubic_function#In_practice | Cubic_function: In_practice.]] {{RoundBoxBottom}} ===Examples=== {{RoundBoxTop|theme=2}} Python function <code>equalRoots()</code> below implements <code>status</code> as presented under [https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above. <syntaxhighlight lang=python> # python code def equalRoots(abcde) : ''' This function returns True if quartic function contains at least 2 equal roots. ''' a,b,c,d,e = abcde aa = a*a ; aaa = aa*a bb = b*b ; bbb = bb*b ; bbbb = bb*bb cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd ee = e*e ; eee = ee*e ; eeee = ee*ee v1 = ( +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd +20*bb*ccc*dd*e ) v2 = ( -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd ) return (v1+v2) == 0 t1 = ( ((1, -1, -19, -11, 30), '4 unique, real roots.'), ((4, 4,-119, -60, 675), '4 unique, real roots, B4 = 0.'), ((1, 6, -48,-182, 735), '2 equal roots.'), ((1,-12, 50, -84, 45), '2 equal roots. B4 = 0.'), ((1,-20, 146,-476, 637), '2 equal roots, 2 complex roots.'), ((1,-12, 58,-132, 117), '2 equal roots, 2 complex roots. B4 = 0.'), ((1, -2, -36, 162, -189), '3 equal roots.'), ((1,-20, 150,-500, 625), '4 equal roots. B4 = 0.'), ((1, -6, -11, 60, 100), '2 pairs of equal roots, B4 = 0.'), ((4, 4, -75,-776,-1869), '2 complex roots.'), ((1,-12, 33, 18, -208), '2 complex roots, B4 = 0.'), ((1,-20, 408,2296,18020), '4 complex roots.'), ((1,-12, 83, -282, 442), '4 complex roots, B4 = 0.'), ((1,-12, 62,-156, 169), '2 pairs of equal complex roots, B4 = 0.'), ) for (abcde, comment) in t1 : print () fourRoots = rootsOfQuartic (abcde) print (comment) print (' Coefficients =', abcde) print (' Four roots =', fourRoots) print (' Equal roots detected:', equalRoots(abcde)) # Check results. a,b,c,d,e = abcde for x in fourRoots : # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0 sum = (a*x**4 + b*x**3 + c*x**2 + d*x + e) if sum : # Create exception. 1/0 </syntaxhighlight> <syntaxhighlight> 4 unique, real roots. Coefficients = (1, -1, -19, -11, 30) Four roots = [5.0, 1.0, -2.0, -3.0] Equal roots detected: False 4 unique, real roots, B4 = 0. Coefficients = (4, 4, -119, -60, 675) Four roots = (2.5, -3.0, 4.5, -5.0) Equal roots detected: False 2 equal roots. Coefficients = (1, 6, -48, -182, 735) Four roots = [5.0, 3.0, -7.0, -7.0] Equal roots detected: True 2 equal roots. B4 = 0. Coefficients = (1, -12, 50, -84, 45) Four roots = (3.0, 3.0, 5.0, 1.0) Equal roots detected: True 2 equal roots, 2 complex roots. Coefficients = (1, -20, 146, -476, 637) Four roots = [7.0, 7.0, (3+2j), (3-2j)] Equal roots detected: True 2 equal roots, 2 complex roots. B4 = 0. Coefficients = (1, -12, 58, -132, 117) Four roots = ((3+2j), (3-2j), 3.0, 3.0) Equal roots detected: True 3 equal roots. Coefficients = (1, -2, -36, 162, -189) Four roots = [3.0, 3.0, 3.0, -7.0] Equal roots detected: True 4 equal roots. B4 = 0. Coefficients = (1, -20, 150, -500, 625) Four roots = (5.0, 5.0, 5.0, 5.0) Equal roots detected: True 2 pairs of equal roots, B4 = 0. Coefficients = (1, -6, -11, 60, 100) Four roots = (5.0, -2.0, 5.0, -2.0) Equal roots detected: True 2 complex roots. Coefficients = (4, 4, -75, -776, -1869) Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)] Equal roots detected: False 2 complex roots, B4 = 0. Coefficients = (1, -12, 33, 18, -208) Four roots = ((3+2j), (3-2j), 8.0, -2.0) Equal roots detected: False 4 complex roots. Coefficients = (1, -20, 408, 2296, 18020) Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)] Equal roots detected: False 4 complex roots, B4 = 0. Coefficients = (1, -12, 83, -282, 442) Four roots = ((3+5j), (3-5j), (3+2j), (3-2j)) Equal roots detected: False 2 pairs of equal complex roots, B4 = 0. Coefficients = (1, -12, 62, -156, 169) Four roots = ((3+2j), (3-2j), (3+2j), (3-2j)) Equal roots detected: True </syntaxhighlight> When description contains note <math>B4 = 0,</math> depressed quartic was processed as quadratic in <math>t^2.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Two Conic Sections= Examples of conic sections include: ellipse, circle, parabola and hyperbola. This section presents examples of two conic sections, circle and ellipse, and how to calculate the coordinates of the point/s of intersection, if any, of the two sections. Let one section with name <math>ABCDEF</math> have equation <math>Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.</math> Let other section with name <math>abcdef</math> have equation <math>ax^2 + by^2 + cxy + dx + ey + f = 0.</math> Because there can be as many as 4 points of intersection, a special "resolvent" quartic function is used to calculate the <math>x</math> coordinates of the point/s of intersection. Coefficients of associated "resolvent" quartic are calculated as follows: <syntaxhighlight lang=python> # python code def intersection_of_2_conic_sections (abcdef, ABCDEF) : ''' A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF) where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function: y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_ ''' A,B,C,D,E,F = ABCDEF a,b,c,d,e,f = abcdef G = ((-1)*(B)*(a) + (1)*(A)*(b)) H = ((-1)*(B)*(d) + (1)*(D)*(b)) I = ((-1)*(B)*(f) + (1)*(F)*(b)) J = ((-1)*(C)*(a) + (1)*(A)*(c)) K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c)) L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c)) M = ((-1)*(E)*(f) + (1)*(F)*(e)) g = ((-1)*(C)*(b) + (1)*(B)*(c)) h = ((-1)*(E)*(b) + (1)*(B)*(e)) A_ = (J)*(g) + (G)*(G) B_ = (J)*(h) + (K)*(g) + (2)*(G)*(H) C_ = (K)*(h) + (L)*(g) + (2)*(G)*(I) + (H)*(H) D_ = (L)*(h) + (M)*(g) + (2)*(H)*(I) E_ = (M)*(h) + (I)*(I) str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_) print (str1) return A_,B_,C_,D_,E_ </syntaxhighlight> == With no common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections06.png|thumb|400px|'''Two conic sections with no common point.''' </br> Resolvent quartic function (black curve) has no real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 80 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.98x^2 + 6850.532x - 22836.7009.</math> <math>f(x)</math> has no real roots. Therefore, there is no point of intersection. {{RoundBoxBottom}} ==With one common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections05.png|thumb|400px|'''Two conic sections with one common point.''' </br> Resolvent quartic function (black curve) has two equal, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 73 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.84x^2 + 7456.48x - 24355.36.</math> Roots of <math>f(x)</math> are: <math>(-14-22.978250586152114j),</math> <math>(-14+22.978250586152114j),</math> <math>5.8, 5.8.</math> <math>f(x)</math> has 2 equal, real roots at <math>x = 5.8,</math> effectively 1 real root where <math>x = 5.8</math> Therefore, there is one point of intersection where <math>x = 5.8.</math> {{RoundBoxBottom}} ==With two common points== ===Example 1=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections04.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two unique, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 6.8x - 17.6y + 64 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.66x^2 + 8235.556x - 26681.1841.</math> Roots of <math>f(x)</math> are: <math>(-14.361578825892241-23.341853011785357j),</math> <math>(-14.361578825892241+23.341853011785357j),</math> <math>4.59885619413921, 7.72430145764527.</math> <math>f(x)</math> has 2 unique, real roots at <math>x = 4.59885619413921, 7.72430145764527.</math> Therefore, there are two points of intersections where <math>x = 4.59885619413921, 7.72430145764527.</math> {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections01.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two pairs of equal roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 18.8x - 1.6y + 53 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 + 37.6x^3 - 504.24x^2 + 2835.04x - 5685.16.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 13, 13.</math> <math>f(x)</math> has 2 pairs of equal roots at <math>x = 5.8, 13,</math> effectively 2 real roots. Therefore, there are two points of intersection where <math>x = 5.8, 13.</math> {{RoundBoxBottom}} ==With 3 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections03.png|thumb|400px|'''Two conic sections with three common points.''' </br> Resolvent quartic function (black curve) has one pair of equal roots and 2 unique, real roots. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 17.6x - 3.2y + 55 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 6.83589838486224, 13.7641016151377 .</math> <math>f(x)</math> has 1 pair of equal roots at <math>x = 5.8</math> and 2 unique, real roots at <math>x = 6.83589838486224, 13.7641016151377,</math> effectively 3 real roots. Therefore, there are three points of intersection where <math>x = 5.8, 6.83589838486224, 13.7641016151377.</math> {{RoundBoxBottom}} ==With 4 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections02.png|thumb|400px|'''Two conic sections with four common points.''' </br> Resolvent quartic function (black curve) has 4 unique, real roots. </br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 18.8x - 1.6y + 62.99 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x)</math><math>= -x^4 + 37.6x^3 - 504.4398x^2 + 2838.79624x - 5544.61147921.</math> Roots of <math>f(x)</math> are:<math>4.36661032156283,</math><math>8.77936456353008,</math> <math>10.0206354364699,</math><math>14.4333896784371.</math> <math>f(x)</math> has 4 real roots as shown above. Therefore, there are four points of intersection where <math>x = 4.36661032156283,</math> <math>8.77936456353008,</math><math>10.0206354364699,</math><math>14.4333896784371.</math> {{RoundBoxBottom}} ===Values of y=== Section above calculates <math>x</math> coordinates of the four common points. This section calculates <math>y</math> coordinates. Here is the lazy way to do it: {{RoundBoxTop|theme=2}} [[File:0818ResQuartic.png|thumb|400px|'''Graphs of 2 curves, red and blue, with coefficients of <math>x,y</math> reversed.''' </br> Roots of "resolvent" quartic, black curve, are interpreted as values of <math>y.</math> ]] <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 # Coefficients of x,y are reversed. abcde = intersection_of_2_conic_sections ((b,a,0,e,d,f), (B,A,C,E,D,F)) values_of_y = rootsOfQuartic (abcde) print (values_of_y) </syntaxhighlight> <syntaxhighlight> [1.6215767432213488, -0.021576743221348723, 5.862095579401657, -4.262095579401658] </syntaxhighlight> By inspection of the diagram above, it is obvious that: <syntaxhighlight lang=python> # python code four_common_points = ( (4.36661032156283, 1.6215767432213488), (8.77936456353009, -4.262095579401658), (10.0206354364699, 5.862095579401657), (14.4333896784372, -0.021576743221348723), ) </syntaxhighlight> Check results: <syntaxhighlight lang=python> # python code def sum_zero (input) : ''' This function calculates sum of all values in input. If sum is very close to 0, this function returns 0. ''' sum = 0 for v in input : sum += v if sum == 0 : return sum tolerance = 1e-14 max = sorted([ abs(v) for v in input ])[-1] if abs(sum) < tolerance*max : return 0 return sum # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 for (x,y) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum1 = sum_zero(values) if sum1 : print ('Error: sum1 =',sum1) values = a*x**2, b*y**2, d*x, e*y, f sum2 = sum_zero(values) if sum2 : print ('Error: sum2 =',sum2) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. Check results with coefficients of <math>x, y</math> reversed: <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = B,A,C,E,D,F # The circle: a,b,c,d,e,f = b,a,c,e,d,f for (y,x) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum3 = sum_zero(values) if sum3 : print ('Error: sum3 =',sum3) values = a*x**2, b*y**2, d*x, e*y, f sum4 = sum_zero(values) if sum4 : print ('Error: sum4 =',sum4) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. {{RoundBoxBottom}} =Links to related topics= {{RoundBoxTop|theme=1}} [[Cubic_function#Cubic_formula | "Cubic formula"]] [[Complex_square_root | "Complex square root" ]] {{RoundBoxBottom}} [[Category:Mathematics]] tpuwv0ocbt3nwkjm3lrqbcehzb8hm67 2844179 2844178 2026-11-03T11:25:58Z ThaniosAkro 2805358 /* Example 1 */ 2844179 wikitext text/x-wiki The quartic function is the bridge between the cubic function and more advanced functions such as the quintic and sextic. =Objective= {{RoundBoxTop|theme=3}} [[file:Books-aj.svg_aj_ashton_01f.png|right|100px]] * Present quartic function and quartic equation. * Introduce the concept of roots of equal absolute value. * Show how to predict and calculate equal roots, techniques that will be useful when applied to higher order functions. * Simplify the depressed quartic. * Show that the quartic equation is effectively solved when at least one root is known. * Present the "resolvent" cubic function. * Show how to derive and use the quartic formula. {{RoundBoxBottom}} =Lesson= {{RoundBoxTop|theme=2}} ==Introduction== The quartic function is the sum of powers of <math>x</math> from <math>0</math> through <math>4</math>: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx^1 + ex^0</math> usually written as: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx + e.</math> If <math>e == 0</math> the function becomes <math>x(ax^3 + bx^2 + cx + d).</math> Within this page we'll say that: * both coefficients <math>a, e</math> must be non-zero, * coefficient <math>a</math> must be positive (simply for our convenience), * all coefficients must be real numbers, accepting that the function may contain complex roots. The quartic equation is the quartic function equated to zero: <math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>. Roots of the function are values of <math>x</math> that satisfy the quartic equation. {{RoundBoxTop|theme=1}} Because the function is "quartic" (maximum power of <math>x</math> is <math>4</math>), the function contains exactly <math>4</math> roots, an even number of complex roots and an even number of real roots. Other combinations of real and complex roots are possible, but they produce complex coefficients. {{RoundBoxBottom}} {{RoundBoxTop|theme=7}} [[File:0312quartic01a.png|thumb|400px|'''Graph of typical quartic function showing minima and maximum.''']] The figure shows a typical quartic function. The function crosses the <math>X</math> axis in 4 different places. The function has 4 roots: <math>(-2,0), (1,0), (5,0), (10,0).</math> This function contains one local minimum, one local maximum and one absolute minimum. There is no absolute maximum. Because the function contains one absolute minimum: * If abs(<math>x</math>) is very large, <math>f(x)</math> is always positive. * If absolute minimum is above <math>X</math> axis, curve does not cross <math>X</math> axis and function contains only complex roots. * There is always at least one point where the curve is parallel to <math>X</math> axis. The curve is never parallel to the <math>Y</math> axis. For any real value of <math>x</math> there is always a real value of <math>y.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0312quartic02.png|thumb|400px|'''When coefficient d is missing, there is a stationary point at x = 0.''']] If coefficient <math>d</math> is missing, the quartic function becomes <math>y = ax^4 + bx^3 + cx^2 + e,</math> and <math>y' = 4ax^3 + 3bx^2 + 2cx = x(4ax^2 + 3bx + 2c).</math> For a stationary point <math>y' = x(4ax^2 + 3bx + 2c) = 0.</math> When coefficient <math>d</math> is missing, there is always a stationary point at <math>x = 0.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} [[File:0319quartic02.png|thumb|400px|'''Graph of quartic function that is quadratic in <math>x^2</math>.'''</br>Because coefficient <math>d</math> is missing, there is a stationary point where <math>x = 0.</math>]] If coefficients <math>b, d</math> are missing, the quartic function becomes a quadratic in <math>x^2.</math> The curve (red line) in diagram has equation: <math>y = f(x) = \frac{x^4 - 13x^2 + 36}{5}</math> The quartic equation may be solved as: <math>X^2 - 13X + 36 = 0</math> where <math>X = x^2</math> or <math>x = \sqrt{X}.</math> <math>X = 4</math> or <math>X = 9.</math> <math>x = \pm 2</math> or <math>x = \pm 3.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0312quartic03.png|thumb|400px| ]] The quartic function may be expressed as <math>x = ay^4 + by^3 + cy^2 + dy + e.</math> Unless otherwise noted, references to "quartic function" on this page refer to function of form <math>y = ax^4 + bx^3 + cx^2 + dx + e.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0318quartic01.png|thumb|400px|'''Graph of quartic function with coefficient a negative.'''</br>There is no absolute minimum.]] Coefficient <math>a</math> may be negative as shown in diagram. As <code>abs</code><math>(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^4.</math> When <code>abs</code><math>(x)</math> is very large, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "quartic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0502quartic01.png|thumb|400px|'''Graph of quartic function with coefficient b missing.''' </br> Sum of roots is <math>0.</math> </br> <math>Y</math>axis compressed for clarity. ]] When sum of roots is <math>0,</math> coefficient <math>b = 0.</math> In the diagram, roots of <math>f(x)</math> are <math>-5, -4, 2, 7.</math> Sum of roots <math>= 0.</math> Therefore coefficient <math>b = 0.</math> {{RoundBoxBottom}} ==Function as product of linear function and cubic== {{RoundBoxTop|theme=3}} [[File:0313quartic01.png|thumb|400px|'''Graphs of quartic function and associated cubic function.''']] When <math>p</math> is a root of the function, the function may be expressed as: <math>(x - p)( Ax^3 + Bx^2 + Cx + D )</math> where <math>A = a;\ B = Ap + b;\ C = Bp + c;\ D = Cp + d.</math> When one real root <math>p</math> is known, the other three roots may be calculated as roots of the cubic function <math>Ax^3 + Bx^2 + Cx + D.</math> In the diagram the quartic function has equation: <math>y = \frac{ x^4 - 23x^3 + 163x^2 - 393x + 252 }{ 48 }.</math> It is known that <math>3</math> is a root of this function. The associated cubic has equation: <math>y = \frac { x^3 - 20x^2 + 103x - 84} { 48 }</math> The 2 curves coincide at points <math>(1, 0),\ (7, 0),\ (12, 0),</math> the three points that are roots of both functions. <math></math> {{RoundBoxBottom}} ==Function defined by 5 points== {{RoundBoxTop|theme=2}} [[File:0313quartic02.png|thumb|400px|'''Figure 1. Quartic function defined by 5 points''' </br>Any 5 points on the curve may be used to define the function.]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 5 unique points on the curve. For example, let us choose the five points: <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code points = (-5,0), (-2,0), (1,0), (3,-6), (6,2) L11 = [] for point in points : x,y = point L11 += [[x*x*x*x, x*x*x, x*x, x, 1, -y]] print (L11) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ 625.0, -125.0, 25.0, -5.0, 1.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 0.0], # [ 1.0, 1.0, 1.0, 1.0, 1.0, 0.0], # matrix supplied to function solveMbyN() below. [ 81.0, 27.0, 9.0, 3.0, 1.0, 6.0], # 5 rows by 6 columns. [1296.0, 216.0, 36.0, 6.0, 1.0, -2.0]] # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L11) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.02651515151515152, 0.004545454545454542, -0.847727272727273, -0.728787878787879, 1.5454545454545459) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by the 5 points <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> is <math>y = \frac{0.875 x^4 + 0.15 x^3 - 27.975 x^2 - 24.05 x + 51}{33}.</math> {{RoundBoxBottom}} ==Function defined by 3 points and 2 slopes== {{RoundBoxTop|theme=2}} [[File:0314quartic03.png|thumb|400px|'''Figure 1. Quartic function defined by 3 points and 2 slopes.''' </br>Any 3 points on the curve and the slopes at any 2 of these points may be used to define the function.</br></br>Slope at <math>(-2, -2)\ =</math> slope at <math>(6, -4)\ =\ 0.</math>]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 3 unique points on the curve and the slopes at any 2 of these points. For example, let us choose the three points: <math>(-2, -2), (6, -4), (4, 1)</math> At point <math>(-2, -2)</math> slope is <math>0.</math> At point <math>(6, -4)</math> slope is <math>0.</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> Rearrange the standard cubic function of slope to prepare for the calculation of <math>a,b,c,d,e:</math> <math> 4 x^3 a + 3 x^2 b + 2 x c + 1 d + 0 e - s = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code def makeEntry(input) : x,y,s = ( tuple(input) + (None,) )[:3] L1 = [] if s != None : L2 = [ float(v) for v in [4*x*x*x, 3*x*x, 2*x, 1, 0, -s] ] L1 += [ L2 ] L2 = [ float(v) for v in [x*x*x*x, x*x*x, x*x, x, 1, -y]] L1 += [ L2 ] return L1 t1 = ( (-2,-2, 0), # point (-2, -2) with slope 0. (6,-4, 0), # point (6, -4) with slope 0. (4,1), # point (4, 1) ) L1 = [] for v in t1 : L1 += makeEntry ( v ) print (L1) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ -32.0, 12.0, -4.0, 1.0, 0.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 2.0], # [ 864.0, 108.0, 12.0, 1.0, 0.0, 0.0], # matrix supplied to function solveMbyN() below. [1296.0, 216.0, 36.0, 6.0, 1.0, 4.0], # 5 rows by 6 columns. [ 256.0, 64.0, 16.0, 4.0, 1.0, -1.0]]. # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L1) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.03255208333333339, -0.2526041666666665, -0.3072916666666667, 2.84375, 2.375) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by three points and two slopes is: <math>y = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> ===Associated cubic functions=== {{RoundBoxTop|theme=3}} ====When p == -2==== {{RoundBoxTop|theme=1}} [[File:0314_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = -2.''' </br>In this case roots of associated cubic include x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == -2,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 15.25 x^2 + 15.75 x + 105 } {48}.</math> Three blue vertical lines show 3 values of <math>x</math> where <math>g(x) = 0</math> and <math>f(x) = f(-2)</math> In this case roots of <math>g(x)</math> include <math>x = p.</math> {{RoundBoxBottom}} ====When p == 5==== {{RoundBoxTop|theme=1}} [[File:0314_2curves02.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 5.''' </br>In this case the one root of associated cubic excludes x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 5,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 4.3125 x^2 - 36.3125 x - 45.0625 } {48}.</math> Two blue vertical lines show 2 values of <math>x</math> where <math>f(x) = f(5)</math> In this case the one root of <math>g(x)</math> excludes <math>x = p.</math> {{RoundBoxBottom}} ====When p == 6==== {{RoundBoxTop|theme=1}} [[File:0315_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 6.''' </br>In this case the one root of associated cubic includes x = p.]] Quartic function is: <math>y = f(x) </math><math>= \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 6,</math> associated cubic function is: <math>y = g(x) </math><math>= \frac{ 1.5625 x^3 - 2.75 x^2 - 31.25 x - 51 } {48}.</math> One blue vertical line shows 1 value of <math>x</math> where <math>f(x) = f(6)</math> In this case the one root of <math>g(x)</math> includes <math>x = p.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} ==Examples== {{RoundBoxTop|theme=2}} ===Quartic with 2 stationary points=== {{RoundBoxTop|theme=1}} [[File:0315_2statPoints02.png|thumb|400px|'''Graph of quartic function with 2 stationary points.''']] In the diagram the red line represents quartic function <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 2 roots of <math>g(x),\ -1</math> and <math>\frac{-1}{4}</math> show that <math>f(x)</math> has stationary points at <math>x = -1</math> and <math>x = -0.25.</math> {{RoundBoxBottom}} ===Quartic with 1 stationary point=== {{RoundBoxTop|theme=1}} [[File:0315_1statPoint.png|thumb|400px|'''Graph of quartic function with 1 stationary point.''']] In the diagram the red line represents quartic function <math>y = f(x) = \frac{x^4 + 3x^3 + 3x^2 + 3x}{2}</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 1 root of <math>g(x),\ -1.607</math> (approx.), shows that <math>f(x)</math> has 1 stationary point where <math>g(x) = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =First and second derivatives= {{RoundBoxTop|theme=1}} ==Points of inflection== {{RoundBoxTop|theme=2}} [[File:0317_3curves01.png|thumb|400px|'''Graphs of quartic function and first two derivatives.'''</br>Dotted portion of black line shows where f(x) is always concave down.</br>Dotted portion of red line shows where g(x) is decreasing.</br>Dotted portion of blue line shows where h(x) is negative.</br>When h(x) is negative, f(x) is concave down.</br>When h(x) is positive, f(x) is concave up.]] In the diagram the black line has equation: <math>y = f(x) = \frac{x^4 - 16x^3 + 42x^2 + 12x + 4}{144}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{x^3 - 12x^2 + 21x + 3}{36}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{x^2 - 8x + 7}{12}.</math> When <math>x < x_1:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. When <math>x == x_1:</math> * <math>y'</math> is at a local maximum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between up and down. When <math>x_1 < x < x_2:</math> * <math>y'</math> is decreasing. * <math>y''</math> is negative. * <math>f(x)</math> is always concave down. When <math>x == x_2:</math> * <math>y'</math> is at a local minimum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between down and up. When <math>x_2 < x:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. {{RoundBoxTop|theme=1}} The roots of <math>h(x): x_1 = 1, x_2 = 7.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At point <math>p_1</math> concavity of <math>f(x)</math> changes from up to down. At point <math>p_2</math> concavity of <math>f(x)</math> changes from down to up. The points <math>p_1, p_2</math> (the <math>X</math> coordinates of which are roots of <math>h(x)</math>) are the '''points of inflection''' of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Maxima and minima== {{RoundBoxTop|theme=2}} [[File:0317_3curves03.png|thumb|400px|'''Graphs of quartic function and first two derivatives showing maximum and minima.'''</br> Point <math>p_{2}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{2}</math> is concave down. Point <math>p_{2}</math> is local maximum. </br> Point <math>p_{1}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{1}</math> is concave up. Point <math>p_{1}</math> is local minimum. </br> Similarly, point <math>p_{3}</math> is local minimum. ]] In the diagram the black line has equation: <math>y = f(x) = \frac{1.5625x^4 - 12.125x^3 - 14.75x^2 + 136.5x + 114}{48}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{6.25x^3 - 36.375x^2 - 29.5x + 136.5}{48}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{18.75x^2 - 72.75x - 29.5}{48}.</math> Roots of <math>g(x):\ x_1 = -2;\ x_2 = 1.82;\ x_3 = 6.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> At <math>x_1\ h(x_1)</math> is positive. Point <math>p_1</math> is a stationary point and <math>f(x)</math> at <math>p_1</math> is concave up. Point <math>p_1</math> is a local minimum. Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At <math>x_2\ h(x_2)</math> is negative. Point <math>p_2</math> is a stationary point and <math>f(x)</math> at <math>p_2</math> is concave down. Point <math>p_2</math> is a local maximum. Let point <math>p_3</math> on <math>f(x)</math> have coordinates <math>(x_3, f(x_3)).</math> At <math>x_3\ h(x_3)</math> is positive. Point <math>p_3</math> is a stationary point and <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is a local minimum. {{RoundBoxBottom}} ==Quartic with 2 stationary points== {{RoundBoxTop|theme=3}} [[File:0318_3curves01.png|thumb|400px|'''Graph of quartic function with 2 stationary points and first 2 derivatives.''' </br> Black line: <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> </br> Red line: <math>y' = g(x) = 4(4x^3 + 9x^2 + 6x + 1).</math> </br> <math>y'' = h(x) = 4(12x^2 + 18x + 6)</math> </br> Blue line: <math>y'' = 4(2x^2 + 3x + 1) = \frac{h(x)}{6}.</math> </br> Dotted portion of black line shows where <math>f(x)</math> is concave down. ]] In the diagram, point <math>p_1</math> on <math>f(x)</math> has coordinates <math>(x_1, f(x_1)).</math> Similarly, points <math>p_2, p_3</math> have coordinates <math>(x_2, f(x_2)),\ (x_3, f(x_3)).</math> <math>y'</math> has roots: <math>x_1 = -1;\ x_3 = -0.25.</math> Points <math>p_1, p_3</math> are stationary points. <math>y''</math> has roots: <math>x_1 = -1;\ x_2 = -0.5.</math> Points <math>p_1, p_2</math> are points of inflection. At point <math>p_3\ y''</math> is positive. <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is local minimum. {{RoundBoxTop|theme=2}} Summary: * Point <math>p_1</math> is both stationary point and point of inflection. * Point <math>p_2</math> is point of inflection. * Point <math>p_3</math> is both stationary point and local minimum. {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =The simplest quartic function= {{RoundBoxTop|theme=2}} [[File:0320quartic01.png|thumb|400px|'''Graph of simplest quartic function.''' </br> Point <math>p_1\ (-1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_2\ (1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_0\ (0, -1.1^4)</math> is <math>Y</math> intercept of <math>f(x).</math> ]] The simplest quartic function has coefficients <math>b = c = d = 0.</math> Red line in diagram has equation: <math>y = f(x) = x^4 - 1.1^4</math> First derivative (not shown) of <math>f(x):\ y' = g(x) = 4x^3.</math> When <math>x == 0,\ g(x) = 0.</math> There is a stationary point on <math>f(x)</math> when <math>x == 0,</math> point <math>p_0.</math> Second derivative (not shown) of <math>f(x):\ y'' = h(x) = 12x^2.</math> When <math>x == 0,\ h(x) = 0.</math> There is a point of inflection on <math>f(x)</math> when <math>x == 0.</math> For every non-zero value of <math>x,\ h(x)</math> is positive. To left and right of point <math>p_0,\ f(x)</math> is always concave up. Point <math>p_0</math> is both local minimum and absolute minimum. * Point <math>p_0</math> is stationary point and point of inflection and absolute minimum. Curve <math>f(x)</math> is useful for finding the fourth root of a real number. Solve: <math>x = N^{\frac{1}{4}}.</math> <math>x^4 = N.</math> <math>x^4 - N = 0.</math> This is equivalent to finding a root of function <math>y = j(x) = x^4 - N.</math> If you use Newton's method to find a root of <math>j(x),</math> this would be more efficient than solving <math>x = \sqrt{\sqrt{N}}.</math> {{RoundBoxBottom}} =Roots of equal absolute value= {{RoundBoxTop|theme=1}} The standard quartic function: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> For <math>x</math> in <math>(1)</math> substitute <math>(p+q).</math> Call this <math>(2).</math> For <math>x</math> in <math>(1)</math> substitute <math>(p-q).</math> Call this <math>(3).</math> Combine <math>(2)</math> and <math>(3)</math> to eliminate <math>q</math> and produce an equation in <math>p:</math> <math>(- 64aaa)pppppp+</math> <math>(- 96aab)ppppp+</math> <math>(- 32aac - 48abb)pppp+</math> <math>(- 32abc - 8bbb)ppp+</math> <math>(+ 16aae - 4abd - 4acc - 8bbc)pp+</math> <math>(+ 8abe - 2bbd - 2bcc)p+</math> <math>(+ add + bbe - bcd)\ =\ 0\ \dots\ (4).</math> We are interested in coefficient <math>0</math> of <math>(4):\ c_0 = add + bbe - bcd.</math> If <math>c_0 == 0,\ p=0</math> is a solution and function <math>(1)</math> has 2 roots of form <math>0 \pm q</math> where <math>q = \sqrt{\frac{-d}{b}}.</math> An example: {{RoundBoxTop|theme=3}} [[File:0320quartic02.png|thumb|400px|'''Graph of quartic function with 2 roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 12x^3 + 31x^2 + 48x -140}{45}.</math> <math>a,b,c,d,e = 1,-12,31,48,-140</math> <math>c_0 = add + bbe - bcd </math><math>= 1(48)(48) + (-12)(-12)(-140) - (-12)(31)(48) = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-48}{-12}} = \sqrt{4} = \pm 2.</math> The 2 roots of equal absolute value are: <math>2, -2.</math> {{RoundBoxBottom}} The method works with complex roots of equal absolute value: {{RoundBoxTop|theme=2}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with 2 complex roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}.</math> <math>a,b,c,d,e = 1,-3,-1,-27,-90</math> <math>c_0 = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-(-27)}{-3}} = \sqrt{-9} = \pm 3i.</math> The 2 roots of equal absolute value are: <math>3i, -3i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Equal roots= {{RoundBoxTop|theme=2}} Equal roots occur when the function and the slope of the function both equal zero. <math>ax^4 + bx^3 + cx^2 + dx + e = 0\ \dots\ (1)</math> <math>4ax^3 + 3bx^2 + 2cx + d = 0\ \dots\ (2)</math> Begin the process of reducing <math>(1),\ (2)</math> to linear functions. Combine <math>(1),\ (2)</math> to produce 2 cubic functions: <math>Fx^3 + Gx^2 + Hx + J\ \dots\ (1a)</math> where: <math>F = ad;\ G = bd - 4ae;\ H = cd - 3be;\ J = dd - 2ce.</math> <math>fx^3 + gx^2 + hx + j\ \dots\ (2a)</math> where: <math>f = 4a;\ g = 3b;\ h = 2c;\ j = d.</math> Combine <math>(1a),\ (2a)</math> to produce 2 quadratic functions: <math>Kx^2 + Lx + M\ \dots\ (1b)</math> where: <math>K = Gf -Fg;\ L = Hf -Fh;\ M = Jf - Fj.</math> <math>kx^2 + lx + m\ \dots\ (2b)</math> where: <math>k = Fj - Jf;\ l = Gj - Jg;\ m = Hj - Jh.</math> Combine <math>(1b),\ (2b)</math> to produce 2 linear functions: <math>Rx + S\ \dots\ (1c)</math> where: <math>R = Lk - Kl;\ S = Mk - Km.</math> <math>rx + s\ \dots\ (2c)</math> where: <math>r = Km-Mk;\ s = Lm - Ml.</math> From <math>(1c):\ x_1 = \frac{-S}{R}</math> From <math>(2c):\ x_2 = \frac{-s}{r}</math> If <math>x_1 == x_2:</math> <math>\frac{-S}{R} = \frac{-s}{r}</math> <math>Rs = rS</math> <math>Rs - Sr = 0.</math> The value <math>Rs - Sr</math> is in fact: <syntaxhighlight> + 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee - 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd + 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee + 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde + 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee - 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde - 3aabbccdddddd </syntaxhighlight> which, by removing values <math>aa, d</math><math>d</math> (common to all values), may be reduced to: <syntaxhighlight> status = ( + 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee - 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd + 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee + 384abccddde - 54abcddddd + 128acccccee - 80accccdde + 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee - 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde - 3bbccdddd ) </syntaxhighlight> If <math>status == 0,</math> there are at least 2 equal roots which may be calculated as shown below. {{RoundBoxTop|theme=2}} If coefficient <math>d</math> is non-zero, it is not necessary to calculate <math>status.</math> If coefficient <math>d == 0,</math> verify that <math>status = 0</math> before proceeding. {{RoundBoxBottom}} ==No equal roots== {{RoundBoxTop|theme=1}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with no equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}</math> <math>a,b,c,d,e = 1, -3, -1, -27, -90</math> <math>R, S = 15269148, -35977608</math> <math>x_1 = \frac{-S}{R} = \frac{35977608}{15269148} = 2.3562289133617\dots</math> <math>r, s = 35977608, -60634332</math> <math>x_2 = \frac{-s}{r} = \frac{60634332}{35977608} = 1.685335278543253\dots</math> <math>x_1 != x_2.</math> There are no equal roots. {{RoundBoxBottom}} ==Exactly 2 equal roots== {{RoundBoxTop|theme=1}} [[File:0329quartic04.png|thumb|400px|'''Graph of quartic function with exactly 2 equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 + 6x^3 - 48x^2 - 182x + 735}{100}</math> <math>a,b,c,d,e = 1, 6, -48, -182, 735</math> <math>R, S = -1027353600, -7191475200</math> <math>x_1 = \frac{-S}{R} = \frac{7191475200}{-1027353600} = -7</math> <math>r, s = 7191475200, 50340326400</math> <math>x_2 = \frac{-s}{r} = \frac{-50340326400}{7191475200} = -7</math> <math>x_1 = x_2 = -7.</math> There are 2 equal roots at <math>x = -7.</math> {{RoundBoxTop|theme=2}} The following 3 graphs show the steps that lead to calculation of equal roots at point <math>(-7, 0).</math> <gallery> File:0914quartic00.png|<small>Figure 1. graph of <math>f(x)</math> and 2 associated cubic functions.</small> File:0914quartic01.png|<small>Figure 2. graph of <math>f(x)</math> and 2 associated quadratic functions.</small> File:0914quartic02.png|<small>Figure 3. graph of <math>f(x)</math> and 2 associated linear functions.</small> </gallery> In all graphs, all curves have a common root at point <math>(-7, 0).</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=4}} See [[Quartic_function#Function_as_product_of_linear_function_and_cubic | "Function as product of linear function and cubic"]] above. To calculate all roots: <syntaxhighlight lang=python> # python code. a,b,c,d,e = 1,6,-48,-182,735 # The associated cubic: p = -7 A = a B = A*p + b C = B*p + c D = C*P + d # The associated quadratic: a1 = A b1 = a1*p + B c1 = b1*p + C a1,b1,c1 </syntaxhighlight> <syntaxhighlight> (1, -8, 15) </syntaxhighlight> Roots of quadratic function <math>g(x) = x^2 - 8x + 15</math> are <math>3, 5.</math> All roots of <math>f(x)</math> are <math>-7, -7, 3, 5.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Exactly 3 equal roots== {{RoundBoxTop|theme=1}} [[File:0320_2curves01.png|thumb|400px|'''Graph of quartic function with exactly 3 equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 2x^3 - 36x^2 + 162x - 189}{100}</math> <math>a,b,c,d,e = 1, -2, -36, 162, -189</math> <math>R, S = 0, 0\ \dots\dots\ (1c)</math> <math>r, s = 0, 0\ \dots\dots\ (2c)</math> In this case the calculation of <math>x_1, x_2</math> is not appropriate because there are more than 2 equal roots. Try equations <math>(1b), (2b).</math> Both of these are equivalent to: <math>y = g(x) = x^2 - 6x + 9,</math> blue line in diagram. Discriminant of <math>g(x) = (-6)^2 - 4(1)(9) = 0.\ g(x)</math> has two equal roots at <math>x = \frac{-(-6)}{2(1)} = 3.</math> Therefore <math>f(x)</math> has 3 equal roots at <math>x = 3.</math> {{RoundBoxBottom}} ==Four equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves01.png|thumb|400px|'''Graph of quartic function with 4 equal roots and corresponding cubic.''' </br> <math>g(x) = \frac{f'(x)}{4}.</math> ]] Red line in diagram is of function: <math>y = f(x) = x^4 - 20x^3 + 150x^2 - 500x + 625.</math> <math>a,b,c,d,e = 1, -20, 150, -500, 625</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> <math>K, L, M = 0, 0, 0</math> <math>k, l, m = 0, 0, 0</math> In this case <math>(1b), (2b), (1c), (2c)</math> are all null. This is the only case in which <math>(1b), (2b)</math> are null. <math>(1a), (2a)</math> are both equivalent to: <math>y = g(x) = x^3 - 15x^2 + 75x - 125,</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 4 equal roots at <math>x = 5.</math> {{RoundBoxBottom}} ==Two pairs of equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 6x^3 - 11x^2 + 60x + 100}{20}.</math> <math>a,b,c,d,e = 1, -6, -11, 60, 100</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> In this case <math>(1c), (2c)</math> are both null. <math>(1b), (2b)</math> are both equivalent to: <math>y = g(x) = \frac{x^2 - 3x - 10}{20},</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = -2</math> and one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 2 equal roots at <math>x = -2</math> and 2 equal roots at <math>x = 5.</math> {{RoundBoxTop|theme=2}} [[File:0408_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal, complex roots and corresponding quadratic.''' </br> <math>g(x) = (x-(3-2i))(x-(3+2i)) = x^2 - 6x + 13.</math> </br> <math>f(x) = (g(x))^2.</math> ]] This method is valid for complex roots. For example: <math>y = f(x) = x^4 - 12x^3 + 62x^2 - 156x + 169.</math> <math>a,b,c,d,e = 1,-12,62,-156,169.</math> In this case <math>(1c),\ (2c)</math> are both null. <math>(1b),\ (2b)</math> are both equivalent to: <math>y = g(x) = x^2 - 6x + 13,</math> blue line in diagram. Roots of <math>g(x)</math> are <math>3 \pm 2i.</math> <math>f(x)</math> has 2 roots equal to <math>3 + 2i</math> and 2 roots equal to <math>3 - 2i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Summmary== {{RoundBoxTop|theme=1}} {| class="wikitable" |- ! || No equal roots !! 2 equal roots !! 3 equal roots !! 4 equal roots !! 2 pairs of equal roots |- | Cubic: 1(a), 2(a) | different | different | different | same | different |- | Quadratic: 1(b), 2(b) | different | different | same, 1root | null | same, 2roots |- | Linear: 1(c), 2(c) | different | same | null | null | null |} {{RoundBoxBottom}} ==Caution== {{RoundBoxTop|theme=4}} [[File:0915quartic02.png|thumb|400px| Calculation of false equal roots.</br> In this example, method calculates 2 legitimate equal roots at <math>(4, 0)</math> and 2 false equal roots at <math>(7.543296089385474,0).</math> <math></math> ]] Black line in diagram has equation: <math>y = f(x) = 0.012684240362811794x^4</math> <math> -\ 0.19522392290249435x^3 </math> <math> +\ 0.7654478458049887x^2 + 0x - 3. </math> <math>f(x)</math> is a quartic function with exactly 2 equal roots and coefficient <math>d</math> missing. Calculation of equal roots of <math>f(x)</math> gives linear functions <math>(1c), (2c)</math> null and quadratic functions <math>(1b), (2b)</math> with equal roots of <math> (4,0), (7.543296089385474,0). </math> Usually, this indicates that <math>f(x)</math> should have 2 equal roots at <math>(4,0)</math> and 2 equal roots at <math>(7.543296089385474,0).</math> It is obvious that <math>7.543296089385474</math> is not a root of <math>f(x).</math> When <math>x = 7.543296089385474,</math> slope of derivative <math>g(x) = 0.</math> Value of <math>f(x)\ != 0.</math> This example indicates that it would be wise to verify that calculated equal roots are in fact valid roots of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Depressed quartic= {{RoundBoxTop|theme=1}} A depressed quartic is any quartic function with any one or more of coefficients <math>b,c,d</math> missing. Within this section a depressed quartic has coefficient <math>b</math> missing. To produce the depressed quartic: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> <math>y = \frac{(4^4 a^3)(ax^4 + bx^3 + cx^2 + dx + e)}{4^4 a^3}\ \dots\ (2)</math> Let <math>x = \frac{-b + t}{4a}.</math> Substitute in <math>(2),</math> expand and simplify: <math>y = \frac{t^4 + At^2 + Bt + C}{4^4a^3}\ \dots\ (3)</math> where: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> When equated to <math>0,\ (3)</math> becomes the depressed equation: <math>t^4 + At^2 + Bt + C = 0\ \dots\ (4).</math> Be prepared for the possibility that any 1 or more of <math>A, B, C</math> may be zero. ==Coefficient B missing== {{RoundBoxTop|theme=2}} [[File:0322_3curves01.png|thumb|400px|'''Graph of quartic function that resembles a quadratic.''' ]] If coefficient <math>B == 0,\ (4)</math> becomes a quadratic in <math>t^2:</math> <math>t^4 + At^2 + C = 0.</math> <math>(1)</math> has the appearance of a quadratic. The black line: <math>y = f(x) = \frac{x^4 - 4x^3 + 9x^2 - 10x + 5}{10}</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>= 8( 8(-10) - 4(-4)(9) + -64 )</math> <math>= 8(-80 + 144 - 64)</math> <math>= 8(0) = 0.</math> The red line: <math>y' = g(x) = \frac{4x^3 - 12x^2 + 18x - 10}{10}</math> <math>y' = g(x) = 0</math> where <math>x = 1.</math> The grey line: <math>y'' = h(x) = \frac{12x^2 - 24x + 18}{10}</math> * Absolute minima of <math>f(x)</math> and of <math>h(x)</math> and point of inflection of <math>g(x)</math> occur where <math>x = \frac{-b}{4} = 1.</math> * <math>y''</math> is always positive. <math>f(x)</math> is always concave up. * <math>f(1 + p) = f(1-p).</math> {{RoundBoxTop|theme=3}} If <math>(1)</math> contains 2 pairs of equal roots, coefficient <math>B = 0.</math> The converse is not necessarily true. If <math>(1)</math> contains 4 equal roots, coefficients <math>A = B = C = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Coefficient C missing== {{RoundBoxTop|theme=2}} [[File:0501quartic01.png|thumb|400px|'''Graph of quartic function with coefficient C of depressed function missing.''' </br> <math>Y</math> axis compressed for clarity. ]] If coefficient <math>C == 0,\ (4)</math> becomes: <math>t^4 + At^2 + Bt = t(t^3 + At + B) = 0</math> in which case <math>t = 0</math> is a solution and <math>x = \frac{-b}{4a}</math> is a root of <math>(1).</math> Curve (red line) in example has equation: <math>y = f(x) = 8x^4 + 16x^3 + 24x^2 + 89x + 40.</math> Coefficients of depressed function are: <syntaxhighlight lang=python> # python code a,b,c,d,e = 8, 16, 24, 89, 40 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b A,B,C </syntaxhighlight> <syntaxhighlight> (1536, 299008, 0) </syntaxhighlight> Coefficient <math>C</math> of depressed function is missing. <math>t = 0</math> is a solution. Using <math>x = \frac{-b + t}{4a},</math> one root of <math>f(x) = \frac{-16 + 0}{4(8)} = -0.5.</math> {{RoundBoxBottom}} ==Resolvent cubic== {{RoundBoxTop|theme=1}} This section introduces a special cubic function called "resolvent" because it helps to resolve a requirement, the calculation of the roots of the quartic. The depressed quartic: <math>t^4 + At^2 + Bt + C\ \dots\ (1)</math> For <math>t</math> substitute <math>(u+v)\ \dots\ (2)</math> For <math>t</math> substitute <math>(u-v)\ \dots\ (3)</math> <math>(2)+(3):\ 2Auu + 2Avv + 2Bu + 2C + 2uuuu + 12uuvv + 2vvvv\ \dots\ (4)</math> Simplify <math>(4):\ Auu + AV + Bu + C + uuuu + 6uuV + VV\ \dots\ (4a)</math> <math>(2)-(3):\ 4Auv + 2Bv + 8uuuv + 8uvvv\ \dots\ (5)</math> Simplify <math>(5):\ 2Au + B + 4uuu + 4uV\ \dots\ (5a)</math> From <math>(5a):\ 4uV = -(2Au + B + 4uuu)\ \dots\ (5b)</math> <math>(4a)*4u4u:\ 4u4uAuu + A4u(4uV) + 4u4uBu + 4u4uC + 4u4uuuuu + 6uu4u(4uV) + (4uV)(4uV)\ \dots\ (6)</math> In <math>(6)</math> replace <math>4uV</math> with <math>(-(2Au + B + 4uuu)),</math> expand, simplify, gather like terms and result is: <math>Pu^6 + Qu^4 + Ru^2 + S</math> or {{RoundBoxTop|theme=2}} <math>PU^3 + QU^2 + RU + S\ \dots\ (7)</math> where: <math>U = u^2</math> <math>P = 64</math> <math>Q = 32A</math> <math>R = 4A^2 - 16C</math> <math>S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} From <math>(5b):\ V = v^2 = \frac{-(2Au + B + 4uuu)}{4u} = -(\frac{A}{2} + U) - \frac{B}{4u}\ \dots\ (8)</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Some simple changes reduce the number of calculations and also the sizes of coefficients <math>P, Q, R, S.</math> <math>A2 = 8ac - 3b^2</math> where <math>A2 = \frac{A}{2}</math> <math>B4 = 16a^2 d - 8abc + 2b^3</math> where <math>B4 = \frac{B}{4}</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> Then: <math>P = 64</math> <math>Q = 32(A2)(2) = 64A2</math> <math>R = 4(A2)(2)(A2)(2) - 16C = 16A2^2 - 16C</math> <math>S = -(B4)(4)(B4)(4) = -16 B4^2</math> Divide all 4 coefficients by <math>16:</math> <math>P = 4</math> <math>Q = 4A2</math> <math>R = A2^2 - C</math> <math>S = -B4^2</math> <math>V = v^2 = -(A2 + U) - \frac{B4}{u}.</math> A close examination of coefficients <math>R, S</math> shows that both coefficients are always exactly divisible by <math>4.</math> Therefore, all four coefficients may be defined as follows: <math>P = 1</math> <math>Q = A2</math> <math>R = \frac{A2^2 - C}{4}</math> <math>S = \frac{-B4^2}{4}</math> Actual calculations of values <math>R, S</math> are: <syntaxhighlight lang=python> # python code. R = ( (16)*(a)*(a)*(b)*(d) + (16)*(a)*(a)*(c)*(c) + (3)*(b)*(b)*(b)*(b) + (-16)*(a)*(b)*(b)*(c) + (-64)*(a)*(a)*(a)*(e) ) S = ( (64)*(a)*(a)*(a)*(b)*(c)*(d) + (8)*(a)*(b)*(b)*(b)*(b)*(c) + (-16)*(a)*(a)*(b)*(b)*(b)*(d) + (-16)*(a)*(a)*(b)*(b)*(c)*(c) + (-1)*(b)*(b)*(b)*(b)*(b)*(b) + (-64)*(a)*(a)*(a)*(a)*(d)*(d) ) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =Solving quartic equation= This section presents 4 examples that show how to use the depressed quartic and the resolvent cubic to solve the quartic equation. ==Four real roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves01.png|thumb|400px|'''Graphs of quartic function with 4 real roots and associated resolvent cubic.''' </br>Resolvent cubic contains 3 real, positive roots. ]] Calculate roots of: <math>y = f(x) = x^4 - x^3 - 19x^2 - 11x + 30</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,-1,-19,-11,30 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -310 -1320 6669 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 -9920 277696 -1742400 </syntaxhighlight> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = 64x^3 - 9920x^2 + 277696x - 1742400.</math> There are 3 real, positive roots: <math>9, 25, 121.</math> Using 3 roots of <math>g(x),</math> calculate 4 roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code for U in (9, 25, 121) : print ('\nU =', U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** .5 for t in (u+v, u-v) : x = (-b+t) / (4*a) y = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x:',x, '; y:',y) </syntaxhighlight> <syntaxhighlight> U = 9 x: 5.0 ; y: 0.0 x: -3.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 U = 25 x: 5.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -3.0 ; y: 0.0 U = 121 x: 5.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: -3.0 ; y: 0.0 </syntaxhighlight> Roots of <math>f(x)</math> are: <math>5, 1, -2, -3.</math> {{RoundBoxBottom}} All 3 values of <math>U</math> produce the same results, but not in same sequence. It is not necessary to calculate all 3 roots of resolvent cubic. Any one non-zero root is sufficient to do the job. ==Two real and two complex roots== ===Example 1=== {{RoundBoxTop|theme=4}} [[File:0323_2curves02.png|thumb|400px|'''Graphs of quartic function with 2 real and 2 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 + 2x^3 + 18x^2 - 70x - 87</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,2,18,-70,-87 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> 264 -5568 -12208 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 8448 474112 -31002624 </syntaxhighlight> {{RoundBoxBottom}} Calculate one real root of cubic function: <math>y = g(x) = 64x^3 + 8448x^2 + 474112x -31002624.</math> <math>36</math> is one real root. Choose <math>U = 36.</math> Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 36 u1 = U**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) if V >= 0 : v = V**.5 else : v = 1j * (-V)**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =',x, '; sum =',sum ) </syntaxhighlight> <syntaxhighlight> x = 3.0 ; sum = 0.0 x = -1.0 ; sum = 0.0 x = (-2+5j) ; sum = 0j x = (-2-5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=4}} [[File:0321_2curves00.png|thumb|400px| Coefficient <math>d</math> of <math>g(x) = 0.</math> <math></math> ]] Calculate roots of: <math>y = f(x)</math> <math>= 3x^4 - 6x^3 - 41x^2 +44x - 189</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 3, -6, -41, 44, -189 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -2184, 0, -1229040 </syntaxhighlight> Notice that coefficient <math>B = 0.</math> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 1, -1092, 605376, 0 </syntaxhighlight> Notice that coefficient <math>S = 0.</math> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = x^3 - 1092x^2 + 605376x + 0.</math> Roots are <math>0, 546 \pm 554.3103823671355j.</math> Value <math>0</math> cannot be used because it will cause error <code>Divide by zero</code> at statement <code>V = -(A/2 + U) - B/(4*u)</code>. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 546+554.3103823671355j print ('\nU =',U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** 0.5 s1 = '\nu,v' print (s1,eval(s1)) for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> U = (546+554.3103823671355j) u,v ((25.729935131257832+10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (4.788322521876306 + 0j) ; sum = (1.9895196601282805e-13 + 0j) x = (0.5 + 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) u,v ((-25.729935131257832-10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (0.5 - 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) x = (-3.7883225218763052 + 0j) ; sum = (1.7053025658242404e-13 + 0j) </syntaxhighlight> Values of <math>x</math> are: <math>-3.7883225218763052, 4.788322521876306, 0.5 \pm 1.795283650280614j</math> <math></math><math></math> {{RoundBoxBottom}} ====Depressed quartic as quadratic==== In this example coefficient <math>B</math> of depressed quartic <math>= 0.</math> Therefore, resolvent cubic can be ignored and depressed quartic processed as quadratic in <math>T = t^2.</math> <math>t^4 - 2184t^2 + (0)t - 1229040</math> <math>T^2 - 2184T - 1229040</math> where <math>T = t^2.</math> Solutions of this quadratic are: <math>T_1, T_2 = 2648.1182474349434, -464.11824743494344</math> <syntaxhighlight lang=python> T1,T2 = 2648.1182474349434, -464.11824743494344 t1 = T1 ** 0.5; t2 = ((-T2) ** 0.5) * 1j for t in (t1,-t1,t2,-t2) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> x = 4.788322521876305 ; sum = -1.7053025658242404e-13 x = -3.788322521876305 ; sum = -1.7053025658242404e-13 x = (0.5 + 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) x = (0.5 - 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) </syntaxhighlight> or <syntaxhighlight> x = 0.5 ± 4.288322521876305, 0.5 ± 1.7952836502806138j </syntaxhighlight> With precision of 15, values of <math>x</math> are same as those shown above. When roots of quartic function are of form <code>p ± q, p ± r,</code> coefficient <math>B</math> of depressed function <math>= 0.</math> ==Four complex roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves03.png|thumb|400px|'''Graphs of quartic function with 4 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 - 20x^3 + 408x^2 + 2296x + 18020</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 4128 344064 9683200 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 64 132096 -86769664 -118380036096 </syntaxhighlight> {{RoundBoxBottom}} Calculate one root of cubic function: <math>y = g(x) = 64x^3 + 132096x^2 - 86769664x - 118380036096.</math> There are 3 real roots: <math>-2304, -784, 1024.</math> Choose <math>U = -784.</math> Negative <math>U</math> is chosen here to show that any 1 of the roots produces the correct result. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = -784 u1 = 1j * (-U)**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) v = V**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> # python expresses complex numbers with 'j'. x = (13+19j) ; sum = 0j x = (-3-5j) ; sum = 0j x = (13-19j) ; sum = 0j x = (-3+5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} =Quartic formula= The substitutions made above can be used to produce a formula for the solution of the quartic equation. See main articles [https://en.wikipedia.org/wiki/Quartic_equation#The_general_case "The general case" ] or [https://en.wikipedia.org/wiki/Quartic_function#General_formula_for_roots "General formula for roots."] {{RoundBoxTop|theme=2}} Both links above point to formula for equation <math>x^4 + ax^3 + bx^2 + cx + d = 0.</math> {{RoundBoxBottom}} Given quartic equation: <math>ax^4 + bx^3 + cx^2 + dx + e = 0,</math> calculate the 4 values of <math>x.</math> <math>x = \frac{-b + t}{4a}</math> where: {{RoundBoxTop|theme=2}} Coefficients of depressed quartic: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of resolvent cubic: <math>a_1 = P = 64</math> <math>b_1 = Q = 32A</math> <math>c_1 = R = 4A^2 - 16C</math> <math>d_1 = S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of depressed cubic: <math>A_1 = 9a_1 c_1 - 3b_1^2</math> <math>B_1 = 27a_1^2d_1 - 9a_1b_1c_1 + 2b_1^3</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of resolvent cubic: <math>C_1 = \frac{-A_1}{3} = b_1^2 - 3a_1 c_1</math> <math>\Delta = B_1^2 - 4C_1^3\ \dots\dots\ \Delta</math> may be negative. <math>\delta = \sqrt{\Delta}</math> <math>W = \frac{-B_1 + \delta}{2}</math> <math>w = \sqrt[3]{W}</math> <math>t_1 = w + \frac{C_1}{w}</math> <math>U = \frac{-b_1 + t_1}{3a_1}</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of quartic: <math>u = \sqrt{U}\ \dots\dots\ u</math> may be positive or negative. <math>V = -(\frac{A}{2} + U) - \frac{B}{4u}</math> <math>v = \sqrt{V}\ \dots\dots\ v</math> may be positive or negative. <math>t = u + v</math> {{RoundBoxBottom}} Formula above produces one value of <math>x.</math> Python code below utilizes <math>\pm \sqrt{U}</math> and <math>\pm \sqrt{V}</math> to produce 4 values of <math>t</math> and then, four values of <math>x.</math> ==An example:== {{RoundBoxTop|theme=4}} [[File:0330quartic01.png|thumb|400px|'''Graph of quartic function with 2 real roots.'''</br><math>Y</math> axis compressed for clarity.]] Calculate roots of <math>f(x) = 4x^4 + 4x^3 - 75x^2 - 776x - 1869.</math> <syntaxhighlight lang=python> # Python code. a,b,c,d,e = 4, 4, -75, -776, -1869 values_of_t = [ t # Coefficients of depressed quartic: for A in (16*a*c - 6*b*b,) for B in (64*a*a*d -32*a*b*c + 8*b*b*b,) for C in (256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b,) # Coefficients of resolvent cubic: for a1 in (64,) for b1 in (32*A,) for c1 in (4*A*A - 16*C,) for d1 in (-B*B,) for U in [ # The resolvent cubic: (-b1+t1)/(3*a1) # Coefficients of depressed resolvent cubic: for A1 in (9*a1*c1 - 3*b1*b1,) for B1 in (27*a1*a1*d1 - 9*a1*b1*c1 + 2*b1*b1*b1,) # One root of resolvent cubic: for C1 in (-A1/3,) for Δ in (B1*B1 - 4*C1*C1*C1,) for δ in (Δ**0.5,) for W in ((-B1 + δ)/2,) for w in (W**(1/3),) for t1 in (w + C1/w,) # See note below. ] # Prepare to calculate 4 values of t. for u1 in (U**.5,) for v1 in ( -(A/2 + U) ,) # Calculate 4 values of t. for u in (u1, -u1,) for V in ( v1 - B/(4*u),) for v in (V**.5,) for t in (u+v, u-v) ] print ('values_of_t =', values_of_t) </syntaxhighlight> {{RoundBoxTop|theme=4}} <syntaxhighlight> values_of_t = [116, -44, (-36+64j), (-36-64j)] </syntaxhighlight> Because <math>f(t)</math> is a depressed quartic function, sum of four <code>values_of_t</code> <math>= 116 - 44 - 36(2) = 0.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> # Python code. # Calculate 4 separate roots. values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] print ('values_of_x =', values_of_x) </syntaxhighlight> <syntaxhighlight lang=python> values_of_x = [7, -3, (-2.5 + 4j), (-2.5 - 4j)] </syntaxhighlight> In python the imaginary part of a complex number is shown with <math>j</math> instead of <math>i.</math> If <math>A == B == C == 0,\ f(x)</math> contains 4 equal roots and <math>x = \frac{-b}{4a}.</math> If <math>f(x)</math> contains 3 or more equal roots, statement <code>for t1 in (w + C1/w,)</code> fails with divisor <math>w = 0.</math> Before using this formula, check for equal roots as in [[Quartic_function#Exactly_3_equal_roots | "Exactly 3 equal roots"]] above. {{RoundBoxTop|theme=1}} Values displayed above have been edited slightly. Actual calculated values were: <syntaxhighlight> values_of_x = [7.000000000000001, -3.0000000000000044, (-2.499999999999998+4.000000000000001j), (-2.4999999999999987-4.000000000000001j)] </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ==In practice== {{RoundBoxTop|theme=5}} The following Python code implements the quartic formula. However, under statement <code>if B4 == 0 :</code> there is code that processes the depressed quartic as a quadratic in <math>T = t^2.</math> This ensures that execution of formula does not fail with error <code>Divide by zero</code> at statement <code>for V in ( v1 - B4/u,).</code> {{RoundBoxTop|theme=4}} <syntaxhighlight lang=python> # python code import cmath cxSqrt = cmath.sqrt # Square root of complex number. def rootsOfQuartic (abcde) : ''' x1,x2,x3,x4 = rootsOfQuartic ((a,b,c,d,e)) Each member of input must be int or float or Decimal object. Int or Decimal object in input is quietly converted to float. Output may be None. ''' def formatResults (x1x2x3x4) : ''' This function improves appearance of results. (8 + 0j) becomes 8.0 ''' values_of_x = list (x1x2x3x4) for p in (0,1,2,3) : v = values_of_x[p] if isinstance (v, complex) and (v.imag == 0) : values_of_x[p] = v.real return values_of_x status = 0 try : a,b,c,d,e = [ float(v) for v in abcde ] except : status = 1 if status : print ('rootsOfQuartic () 1: Error creating coefficients a,b,c,d,e.') return None if a == 0 : print ('rootsOfQuartic () 2: Coefficient a must be non-zero.') return None # Coefficients of depressed quartic, modified. # A = 16*a*c - 6*b*b A2 = 8*a*c - 3*b*b # B = 64*a*a*d - 32*a*b*c + 8*b*b*b B4 = 16*a*a*d - 8*a*b*c + 2*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b if B4 == 0 : # B = 0. # Result returned from this section is type tuple, indicating that coefficient B4 = 0. if A2==C==0 : # 4 equal roots. root = -b/(4*a) return tuple(formatResults((root,root,root,root))) # t**4 + At**2 + (0)t + C # Depressed quartic is quadratic in T: # T**2 + AT + C where T = t**2 # T**2 + 2(A2)T + C where A = 2(A2) # -2(A2) +/- (4(A2)(A2) - 4C)**0.5 # T = -------------------------------- = -A2 +/- ((A2)(A2) - C)**0.5 # 2 disc = A2*A2 - C if disc >= 0 : root = disc ** 0.5 else : root = ((-disc) ** 0.5) * 1j T1 = -A2 - root ; T2 = -A2 + root t1 = cxSqrt(T1) ; t2 = cxSqrt(T2) values_of_t = (t1,-t1,t2,-t2) values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] return tuple(formatResults(values_of_x)) # B4 is non-zero. Therefore, all of (S, U, u) are non-zero. P,Q,R,S = 1, A2, (A2*A2 - C)/4, -B4*B4/4 # str1 = 'P,Q,R,S' ; print (str1, eval(str1)) U = oneRootOfCubic((P,Q,R,S)) # Resolvent cubic. if U > 0 : sqrtU = U ** 0.5 elif U == 0 : # This should not happen. print ('rootsOfQuartic () 3: Internal error.') return None else : sqrtU = ((-U) ** 0.5) * 1j v1 = -(A2+U) values_of_t = [ t for u in (sqrtU, -sqrtU) for V in ( v1 - B4/u, ) for v in ( cxSqrt(V), ) for t in (u+v, u-v) ] values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] # Result returned from this section is type list, indicating that coefficient B4 != 0. return formatResults(values_of_x) </syntaxhighlight> For function <code>oneRootOfCubic()</code> see [[Cubic_function#In_practice | Cubic_function: In_practice.]] {{RoundBoxBottom}} ===Examples=== {{RoundBoxTop|theme=2}} Python function <code>equalRoots()</code> below implements <code>status</code> as presented under [https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above. <syntaxhighlight lang=python> # python code def equalRoots(abcde) : ''' This function returns True if quartic function contains at least 2 equal roots. ''' a,b,c,d,e = abcde aa = a*a ; aaa = aa*a bb = b*b ; bbb = bb*b ; bbbb = bb*bb cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd ee = e*e ; eee = ee*e ; eeee = ee*ee v1 = ( +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd +20*bb*ccc*dd*e ) v2 = ( -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd ) return (v1+v2) == 0 t1 = ( ((1, -1, -19, -11, 30), '4 unique, real roots.'), ((4, 4,-119, -60, 675), '4 unique, real roots, B4 = 0.'), ((1, 6, -48,-182, 735), '2 equal roots.'), ((1,-12, 50, -84, 45), '2 equal roots. B4 = 0.'), ((1,-20, 146,-476, 637), '2 equal roots, 2 complex roots.'), ((1,-12, 58,-132, 117), '2 equal roots, 2 complex roots. B4 = 0.'), ((1, -2, -36, 162, -189), '3 equal roots.'), ((1,-20, 150,-500, 625), '4 equal roots. B4 = 0.'), ((1, -6, -11, 60, 100), '2 pairs of equal roots, B4 = 0.'), ((4, 4, -75,-776,-1869), '2 complex roots.'), ((1,-12, 33, 18, -208), '2 complex roots, B4 = 0.'), ((1,-20, 408,2296,18020), '4 complex roots.'), ((1,-12, 83, -282, 442), '4 complex roots, B4 = 0.'), ((1,-12, 62,-156, 169), '2 pairs of equal complex roots, B4 = 0.'), ) for (abcde, comment) in t1 : print () fourRoots = rootsOfQuartic (abcde) print (comment) print (' Coefficients =', abcde) print (' Four roots =', fourRoots) print (' Equal roots detected:', equalRoots(abcde)) # Check results. a,b,c,d,e = abcde for x in fourRoots : # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0 sum = (a*x**4 + b*x**3 + c*x**2 + d*x + e) if sum : # Create exception. 1/0 </syntaxhighlight> <syntaxhighlight> 4 unique, real roots. Coefficients = (1, -1, -19, -11, 30) Four roots = [5.0, 1.0, -2.0, -3.0] Equal roots detected: False 4 unique, real roots, B4 = 0. Coefficients = (4, 4, -119, -60, 675) Four roots = (2.5, -3.0, 4.5, -5.0) Equal roots detected: False 2 equal roots. Coefficients = (1, 6, -48, -182, 735) Four roots = [5.0, 3.0, -7.0, -7.0] Equal roots detected: True 2 equal roots. B4 = 0. Coefficients = (1, -12, 50, -84, 45) Four roots = (3.0, 3.0, 5.0, 1.0) Equal roots detected: True 2 equal roots, 2 complex roots. Coefficients = (1, -20, 146, -476, 637) Four roots = [7.0, 7.0, (3+2j), (3-2j)] Equal roots detected: True 2 equal roots, 2 complex roots. B4 = 0. Coefficients = (1, -12, 58, -132, 117) Four roots = ((3+2j), (3-2j), 3.0, 3.0) Equal roots detected: True 3 equal roots. Coefficients = (1, -2, -36, 162, -189) Four roots = [3.0, 3.0, 3.0, -7.0] Equal roots detected: True 4 equal roots. B4 = 0. Coefficients = (1, -20, 150, -500, 625) Four roots = (5.0, 5.0, 5.0, 5.0) Equal roots detected: True 2 pairs of equal roots, B4 = 0. Coefficients = (1, -6, -11, 60, 100) Four roots = (5.0, -2.0, 5.0, -2.0) Equal roots detected: True 2 complex roots. Coefficients = (4, 4, -75, -776, -1869) Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)] Equal roots detected: False 2 complex roots, B4 = 0. Coefficients = (1, -12, 33, 18, -208) Four roots = ((3+2j), (3-2j), 8.0, -2.0) Equal roots detected: False 4 complex roots. Coefficients = (1, -20, 408, 2296, 18020) Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)] Equal roots detected: False 4 complex roots, B4 = 0. Coefficients = (1, -12, 83, -282, 442) Four roots = ((3+5j), (3-5j), (3+2j), (3-2j)) Equal roots detected: False 2 pairs of equal complex roots, B4 = 0. Coefficients = (1, -12, 62, -156, 169) Four roots = ((3+2j), (3-2j), (3+2j), (3-2j)) Equal roots detected: True </syntaxhighlight> When description contains note <math>B4 = 0,</math> depressed quartic was processed as quadratic in <math>t^2.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Two Conic Sections= Examples of conic sections include: ellipse, circle, parabola and hyperbola. This section presents examples of two conic sections, circle and ellipse, and how to calculate the coordinates of the point/s of intersection, if any, of the two sections. Let one section with name <math>ABCDEF</math> have equation <math>Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.</math> Let other section with name <math>abcdef</math> have equation <math>ax^2 + by^2 + cxy + dx + ey + f = 0.</math> Because there can be as many as 4 points of intersection, a special "resolvent" quartic function is used to calculate the <math>x</math> coordinates of the point/s of intersection. Coefficients of associated "resolvent" quartic are calculated as follows: <syntaxhighlight lang=python> # python code def intersection_of_2_conic_sections (abcdef, ABCDEF) : ''' A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF) where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function: y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_ ''' A,B,C,D,E,F = ABCDEF a,b,c,d,e,f = abcdef G = ((-1)*(B)*(a) + (1)*(A)*(b)) H = ((-1)*(B)*(d) + (1)*(D)*(b)) I = ((-1)*(B)*(f) + (1)*(F)*(b)) J = ((-1)*(C)*(a) + (1)*(A)*(c)) K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c)) L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c)) M = ((-1)*(E)*(f) + (1)*(F)*(e)) g = ((-1)*(C)*(b) + (1)*(B)*(c)) h = ((-1)*(E)*(b) + (1)*(B)*(e)) A_ = (J)*(g) + (G)*(G) B_ = (J)*(h) + (K)*(g) + (2)*(G)*(H) C_ = (K)*(h) + (L)*(g) + (2)*(G)*(I) + (H)*(H) D_ = (L)*(h) + (M)*(g) + (2)*(H)*(I) E_ = (M)*(h) + (I)*(I) str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_) print (str1) return A_,B_,C_,D_,E_ </syntaxhighlight> == With no common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections06.png|thumb|400px|'''Two conic sections with no common point.''' </br> Resolvent quartic function (black curve) has no real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 80 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.98x^2 + 6850.532x - 22836.7009.</math> <math>f(x)</math> has no real roots. Therefore, there is no point of intersection. {{RoundBoxBottom}} ==With one common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections05.png|thumb|400px|'''Two conic sections with one common point.''' </br> Resolvent quartic function (black curve) has two equal, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 73 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.84x^2 + 7456.48x - 24355.36.</math> Roots of <math>f(x)</math> are: <math>(-14-22.978250586152114j),</math> <math>(-14+22.978250586152114j),</math> <math>5.8, 5.8.</math> <math>f(x)</math> has 2 equal, real roots at <math>x = 5.8,</math> effectively 1 real root where <math>x = 5.8</math> Therefore, there is one point of intersection where <math>x = 5.8.</math> {{RoundBoxBottom}} ==With two common points== ===Example 1=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections04.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two unique, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 64 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.66x^2 + 8235.556x - 26681.1841.</math> Roots of <math>f(x)</math> are: <math>(-14.361578825892241-23.341853011785357j),</math> <math>(-14.361578825892241+23.341853011785357j),</math> <math>4.59885619413921, 7.72430145764527.</math> <math>f(x)</math> has 2 unique, real roots at <math>x = 4.59885619413921, 7.72430145764527.</math> Therefore, there are two points of intersections where <math>x = 4.59885619413921, 7.72430145764527.</math> {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections01.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two pairs of equal roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 18.8x - 1.6y + 53 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 + 37.6x^3 - 504.24x^2 + 2835.04x - 5685.16.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 13, 13.</math> <math>f(x)</math> has 2 pairs of equal roots at <math>x = 5.8, 13,</math> effectively 2 real roots. Therefore, there are two points of intersection where <math>x = 5.8, 13.</math> {{RoundBoxBottom}} ==With 3 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections03.png|thumb|400px|'''Two conic sections with three common points.''' </br> Resolvent quartic function (black curve) has one pair of equal roots and 2 unique, real roots. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 17.6x - 3.2y + 55 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 6.83589838486224, 13.7641016151377 .</math> <math>f(x)</math> has 1 pair of equal roots at <math>x = 5.8</math> and 2 unique, real roots at <math>x = 6.83589838486224, 13.7641016151377,</math> effectively 3 real roots. Therefore, there are three points of intersection where <math>x = 5.8, 6.83589838486224, 13.7641016151377.</math> {{RoundBoxBottom}} ==With 4 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections02.png|thumb|400px|'''Two conic sections with four common points.''' </br> Resolvent quartic function (black curve) has 4 unique, real roots. </br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 18.8x - 1.6y + 62.99 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x)</math><math>= -x^4 + 37.6x^3 - 504.4398x^2 + 2838.79624x - 5544.61147921.</math> Roots of <math>f(x)</math> are:<math>4.36661032156283,</math><math>8.77936456353008,</math> <math>10.0206354364699,</math><math>14.4333896784371.</math> <math>f(x)</math> has 4 real roots as shown above. Therefore, there are four points of intersection where <math>x = 4.36661032156283,</math> <math>8.77936456353008,</math><math>10.0206354364699,</math><math>14.4333896784371.</math> {{RoundBoxBottom}} ===Values of y=== Section above calculates <math>x</math> coordinates of the four common points. This section calculates <math>y</math> coordinates. Here is the lazy way to do it: {{RoundBoxTop|theme=2}} [[File:0818ResQuartic.png|thumb|400px|'''Graphs of 2 curves, red and blue, with coefficients of <math>x,y</math> reversed.''' </br> Roots of "resolvent" quartic, black curve, are interpreted as values of <math>y.</math> ]] <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 # Coefficients of x,y are reversed. abcde = intersection_of_2_conic_sections ((b,a,0,e,d,f), (B,A,C,E,D,F)) values_of_y = rootsOfQuartic (abcde) print (values_of_y) </syntaxhighlight> <syntaxhighlight> [1.6215767432213488, -0.021576743221348723, 5.862095579401657, -4.262095579401658] </syntaxhighlight> By inspection of the diagram above, it is obvious that: <syntaxhighlight lang=python> # python code four_common_points = ( (4.36661032156283, 1.6215767432213488), (8.77936456353009, -4.262095579401658), (10.0206354364699, 5.862095579401657), (14.4333896784372, -0.021576743221348723), ) </syntaxhighlight> Check results: <syntaxhighlight lang=python> # python code def sum_zero (input) : ''' This function calculates sum of all values in input. If sum is very close to 0, this function returns 0. ''' sum = 0 for v in input : sum += v if sum == 0 : return sum tolerance = 1e-14 max = sorted([ abs(v) for v in input ])[-1] if abs(sum) < tolerance*max : return 0 return sum # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 for (x,y) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum1 = sum_zero(values) if sum1 : print ('Error: sum1 =',sum1) values = a*x**2, b*y**2, d*x, e*y, f sum2 = sum_zero(values) if sum2 : print ('Error: sum2 =',sum2) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. Check results with coefficients of <math>x, y</math> reversed: <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = B,A,C,E,D,F # The circle: a,b,c,d,e,f = b,a,c,e,d,f for (y,x) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum3 = sum_zero(values) if sum3 : print ('Error: sum3 =',sum3) values = a*x**2, b*y**2, d*x, e*y, f sum4 = sum_zero(values) if sum4 : print ('Error: sum4 =',sum4) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. {{RoundBoxBottom}} =Links to related topics= {{RoundBoxTop|theme=1}} [[Cubic_function#Cubic_formula | "Cubic formula"]] [[Complex_square_root | "Complex square root" ]] {{RoundBoxBottom}} [[Category:Mathematics]] 2evq8320f5aj0udbnk6xqewaak782s0 2844180 2844179 2026-11-03T11:27:28Z ThaniosAkro 2805358 /* Example 2 */ 2844180 wikitext text/x-wiki The quartic function is the bridge between the cubic function and more advanced functions such as the quintic and sextic. =Objective= {{RoundBoxTop|theme=3}} [[file:Books-aj.svg_aj_ashton_01f.png|right|100px]] * Present quartic function and quartic equation. * Introduce the concept of roots of equal absolute value. * Show how to predict and calculate equal roots, techniques that will be useful when applied to higher order functions. * Simplify the depressed quartic. * Show that the quartic equation is effectively solved when at least one root is known. * Present the "resolvent" cubic function. * Show how to derive and use the quartic formula. {{RoundBoxBottom}} =Lesson= {{RoundBoxTop|theme=2}} ==Introduction== The quartic function is the sum of powers of <math>x</math> from <math>0</math> through <math>4</math>: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx^1 + ex^0</math> usually written as: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx + e.</math> If <math>e == 0</math> the function becomes <math>x(ax^3 + bx^2 + cx + d).</math> Within this page we'll say that: * both coefficients <math>a, e</math> must be non-zero, * coefficient <math>a</math> must be positive (simply for our convenience), * all coefficients must be real numbers, accepting that the function may contain complex roots. The quartic equation is the quartic function equated to zero: <math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>. Roots of the function are values of <math>x</math> that satisfy the quartic equation. {{RoundBoxTop|theme=1}} Because the function is "quartic" (maximum power of <math>x</math> is <math>4</math>), the function contains exactly <math>4</math> roots, an even number of complex roots and an even number of real roots. Other combinations of real and complex roots are possible, but they produce complex coefficients. {{RoundBoxBottom}} {{RoundBoxTop|theme=7}} [[File:0312quartic01a.png|thumb|400px|'''Graph of typical quartic function showing minima and maximum.''']] The figure shows a typical quartic function. The function crosses the <math>X</math> axis in 4 different places. The function has 4 roots: <math>(-2,0), (1,0), (5,0), (10,0).</math> This function contains one local minimum, one local maximum and one absolute minimum. There is no absolute maximum. Because the function contains one absolute minimum: * If abs(<math>x</math>) is very large, <math>f(x)</math> is always positive. * If absolute minimum is above <math>X</math> axis, curve does not cross <math>X</math> axis and function contains only complex roots. * There is always at least one point where the curve is parallel to <math>X</math> axis. The curve is never parallel to the <math>Y</math> axis. For any real value of <math>x</math> there is always a real value of <math>y.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0312quartic02.png|thumb|400px|'''When coefficient d is missing, there is a stationary point at x = 0.''']] If coefficient <math>d</math> is missing, the quartic function becomes <math>y = ax^4 + bx^3 + cx^2 + e,</math> and <math>y' = 4ax^3 + 3bx^2 + 2cx = x(4ax^2 + 3bx + 2c).</math> For a stationary point <math>y' = x(4ax^2 + 3bx + 2c) = 0.</math> When coefficient <math>d</math> is missing, there is always a stationary point at <math>x = 0.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} [[File:0319quartic02.png|thumb|400px|'''Graph of quartic function that is quadratic in <math>x^2</math>.'''</br>Because coefficient <math>d</math> is missing, there is a stationary point where <math>x = 0.</math>]] If coefficients <math>b, d</math> are missing, the quartic function becomes a quadratic in <math>x^2.</math> The curve (red line) in diagram has equation: <math>y = f(x) = \frac{x^4 - 13x^2 + 36}{5}</math> The quartic equation may be solved as: <math>X^2 - 13X + 36 = 0</math> where <math>X = x^2</math> or <math>x = \sqrt{X}.</math> <math>X = 4</math> or <math>X = 9.</math> <math>x = \pm 2</math> or <math>x = \pm 3.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0312quartic03.png|thumb|400px| ]] The quartic function may be expressed as <math>x = ay^4 + by^3 + cy^2 + dy + e.</math> Unless otherwise noted, references to "quartic function" on this page refer to function of form <math>y = ax^4 + bx^3 + cx^2 + dx + e.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0318quartic01.png|thumb|400px|'''Graph of quartic function with coefficient a negative.'''</br>There is no absolute minimum.]] Coefficient <math>a</math> may be negative as shown in diagram. As <code>abs</code><math>(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^4.</math> When <code>abs</code><math>(x)</math> is very large, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "quartic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0502quartic01.png|thumb|400px|'''Graph of quartic function with coefficient b missing.''' </br> Sum of roots is <math>0.</math> </br> <math>Y</math>axis compressed for clarity. ]] When sum of roots is <math>0,</math> coefficient <math>b = 0.</math> In the diagram, roots of <math>f(x)</math> are <math>-5, -4, 2, 7.</math> Sum of roots <math>= 0.</math> Therefore coefficient <math>b = 0.</math> {{RoundBoxBottom}} ==Function as product of linear function and cubic== {{RoundBoxTop|theme=3}} [[File:0313quartic01.png|thumb|400px|'''Graphs of quartic function and associated cubic function.''']] When <math>p</math> is a root of the function, the function may be expressed as: <math>(x - p)( Ax^3 + Bx^2 + Cx + D )</math> where <math>A = a;\ B = Ap + b;\ C = Bp + c;\ D = Cp + d.</math> When one real root <math>p</math> is known, the other three roots may be calculated as roots of the cubic function <math>Ax^3 + Bx^2 + Cx + D.</math> In the diagram the quartic function has equation: <math>y = \frac{ x^4 - 23x^3 + 163x^2 - 393x + 252 }{ 48 }.</math> It is known that <math>3</math> is a root of this function. The associated cubic has equation: <math>y = \frac { x^3 - 20x^2 + 103x - 84} { 48 }</math> The 2 curves coincide at points <math>(1, 0),\ (7, 0),\ (12, 0),</math> the three points that are roots of both functions. <math></math> {{RoundBoxBottom}} ==Function defined by 5 points== {{RoundBoxTop|theme=2}} [[File:0313quartic02.png|thumb|400px|'''Figure 1. Quartic function defined by 5 points''' </br>Any 5 points on the curve may be used to define the function.]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 5 unique points on the curve. For example, let us choose the five points: <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code points = (-5,0), (-2,0), (1,0), (3,-6), (6,2) L11 = [] for point in points : x,y = point L11 += [[x*x*x*x, x*x*x, x*x, x, 1, -y]] print (L11) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ 625.0, -125.0, 25.0, -5.0, 1.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 0.0], # [ 1.0, 1.0, 1.0, 1.0, 1.0, 0.0], # matrix supplied to function solveMbyN() below. [ 81.0, 27.0, 9.0, 3.0, 1.0, 6.0], # 5 rows by 6 columns. [1296.0, 216.0, 36.0, 6.0, 1.0, -2.0]] # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L11) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.02651515151515152, 0.004545454545454542, -0.847727272727273, -0.728787878787879, 1.5454545454545459) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by the 5 points <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> is <math>y = \frac{0.875 x^4 + 0.15 x^3 - 27.975 x^2 - 24.05 x + 51}{33}.</math> {{RoundBoxBottom}} ==Function defined by 3 points and 2 slopes== {{RoundBoxTop|theme=2}} [[File:0314quartic03.png|thumb|400px|'''Figure 1. Quartic function defined by 3 points and 2 slopes.''' </br>Any 3 points on the curve and the slopes at any 2 of these points may be used to define the function.</br></br>Slope at <math>(-2, -2)\ =</math> slope at <math>(6, -4)\ =\ 0.</math>]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 3 unique points on the curve and the slopes at any 2 of these points. For example, let us choose the three points: <math>(-2, -2), (6, -4), (4, 1)</math> At point <math>(-2, -2)</math> slope is <math>0.</math> At point <math>(6, -4)</math> slope is <math>0.</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> Rearrange the standard cubic function of slope to prepare for the calculation of <math>a,b,c,d,e:</math> <math> 4 x^3 a + 3 x^2 b + 2 x c + 1 d + 0 e - s = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code def makeEntry(input) : x,y,s = ( tuple(input) + (None,) )[:3] L1 = [] if s != None : L2 = [ float(v) for v in [4*x*x*x, 3*x*x, 2*x, 1, 0, -s] ] L1 += [ L2 ] L2 = [ float(v) for v in [x*x*x*x, x*x*x, x*x, x, 1, -y]] L1 += [ L2 ] return L1 t1 = ( (-2,-2, 0), # point (-2, -2) with slope 0. (6,-4, 0), # point (6, -4) with slope 0. (4,1), # point (4, 1) ) L1 = [] for v in t1 : L1 += makeEntry ( v ) print (L1) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ -32.0, 12.0, -4.0, 1.0, 0.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 2.0], # [ 864.0, 108.0, 12.0, 1.0, 0.0, 0.0], # matrix supplied to function solveMbyN() below. [1296.0, 216.0, 36.0, 6.0, 1.0, 4.0], # 5 rows by 6 columns. [ 256.0, 64.0, 16.0, 4.0, 1.0, -1.0]]. # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L1) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.03255208333333339, -0.2526041666666665, -0.3072916666666667, 2.84375, 2.375) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by three points and two slopes is: <math>y = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> ===Associated cubic functions=== {{RoundBoxTop|theme=3}} ====When p == -2==== {{RoundBoxTop|theme=1}} [[File:0314_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = -2.''' </br>In this case roots of associated cubic include x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == -2,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 15.25 x^2 + 15.75 x + 105 } {48}.</math> Three blue vertical lines show 3 values of <math>x</math> where <math>g(x) = 0</math> and <math>f(x) = f(-2)</math> In this case roots of <math>g(x)</math> include <math>x = p.</math> {{RoundBoxBottom}} ====When p == 5==== {{RoundBoxTop|theme=1}} [[File:0314_2curves02.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 5.''' </br>In this case the one root of associated cubic excludes x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 5,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 4.3125 x^2 - 36.3125 x - 45.0625 } {48}.</math> Two blue vertical lines show 2 values of <math>x</math> where <math>f(x) = f(5)</math> In this case the one root of <math>g(x)</math> excludes <math>x = p.</math> {{RoundBoxBottom}} ====When p == 6==== {{RoundBoxTop|theme=1}} [[File:0315_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 6.''' </br>In this case the one root of associated cubic includes x = p.]] Quartic function is: <math>y = f(x) </math><math>= \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 6,</math> associated cubic function is: <math>y = g(x) </math><math>= \frac{ 1.5625 x^3 - 2.75 x^2 - 31.25 x - 51 } {48}.</math> One blue vertical line shows 1 value of <math>x</math> where <math>f(x) = f(6)</math> In this case the one root of <math>g(x)</math> includes <math>x = p.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} ==Examples== {{RoundBoxTop|theme=2}} ===Quartic with 2 stationary points=== {{RoundBoxTop|theme=1}} [[File:0315_2statPoints02.png|thumb|400px|'''Graph of quartic function with 2 stationary points.''']] In the diagram the red line represents quartic function <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 2 roots of <math>g(x),\ -1</math> and <math>\frac{-1}{4}</math> show that <math>f(x)</math> has stationary points at <math>x = -1</math> and <math>x = -0.25.</math> {{RoundBoxBottom}} ===Quartic with 1 stationary point=== {{RoundBoxTop|theme=1}} [[File:0315_1statPoint.png|thumb|400px|'''Graph of quartic function with 1 stationary point.''']] In the diagram the red line represents quartic function <math>y = f(x) = \frac{x^4 + 3x^3 + 3x^2 + 3x}{2}</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 1 root of <math>g(x),\ -1.607</math> (approx.), shows that <math>f(x)</math> has 1 stationary point where <math>g(x) = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =First and second derivatives= {{RoundBoxTop|theme=1}} ==Points of inflection== {{RoundBoxTop|theme=2}} [[File:0317_3curves01.png|thumb|400px|'''Graphs of quartic function and first two derivatives.'''</br>Dotted portion of black line shows where f(x) is always concave down.</br>Dotted portion of red line shows where g(x) is decreasing.</br>Dotted portion of blue line shows where h(x) is negative.</br>When h(x) is negative, f(x) is concave down.</br>When h(x) is positive, f(x) is concave up.]] In the diagram the black line has equation: <math>y = f(x) = \frac{x^4 - 16x^3 + 42x^2 + 12x + 4}{144}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{x^3 - 12x^2 + 21x + 3}{36}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{x^2 - 8x + 7}{12}.</math> When <math>x < x_1:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. When <math>x == x_1:</math> * <math>y'</math> is at a local maximum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between up and down. When <math>x_1 < x < x_2:</math> * <math>y'</math> is decreasing. * <math>y''</math> is negative. * <math>f(x)</math> is always concave down. When <math>x == x_2:</math> * <math>y'</math> is at a local minimum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between down and up. When <math>x_2 < x:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. {{RoundBoxTop|theme=1}} The roots of <math>h(x): x_1 = 1, x_2 = 7.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At point <math>p_1</math> concavity of <math>f(x)</math> changes from up to down. At point <math>p_2</math> concavity of <math>f(x)</math> changes from down to up. The points <math>p_1, p_2</math> (the <math>X</math> coordinates of which are roots of <math>h(x)</math>) are the '''points of inflection''' of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Maxima and minima== {{RoundBoxTop|theme=2}} [[File:0317_3curves03.png|thumb|400px|'''Graphs of quartic function and first two derivatives showing maximum and minima.'''</br> Point <math>p_{2}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{2}</math> is concave down. Point <math>p_{2}</math> is local maximum. </br> Point <math>p_{1}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{1}</math> is concave up. Point <math>p_{1}</math> is local minimum. </br> Similarly, point <math>p_{3}</math> is local minimum. ]] In the diagram the black line has equation: <math>y = f(x) = \frac{1.5625x^4 - 12.125x^3 - 14.75x^2 + 136.5x + 114}{48}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{6.25x^3 - 36.375x^2 - 29.5x + 136.5}{48}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{18.75x^2 - 72.75x - 29.5}{48}.</math> Roots of <math>g(x):\ x_1 = -2;\ x_2 = 1.82;\ x_3 = 6.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> At <math>x_1\ h(x_1)</math> is positive. Point <math>p_1</math> is a stationary point and <math>f(x)</math> at <math>p_1</math> is concave up. Point <math>p_1</math> is a local minimum. Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At <math>x_2\ h(x_2)</math> is negative. Point <math>p_2</math> is a stationary point and <math>f(x)</math> at <math>p_2</math> is concave down. Point <math>p_2</math> is a local maximum. Let point <math>p_3</math> on <math>f(x)</math> have coordinates <math>(x_3, f(x_3)).</math> At <math>x_3\ h(x_3)</math> is positive. Point <math>p_3</math> is a stationary point and <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is a local minimum. {{RoundBoxBottom}} ==Quartic with 2 stationary points== {{RoundBoxTop|theme=3}} [[File:0318_3curves01.png|thumb|400px|'''Graph of quartic function with 2 stationary points and first 2 derivatives.''' </br> Black line: <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> </br> Red line: <math>y' = g(x) = 4(4x^3 + 9x^2 + 6x + 1).</math> </br> <math>y'' = h(x) = 4(12x^2 + 18x + 6)</math> </br> Blue line: <math>y'' = 4(2x^2 + 3x + 1) = \frac{h(x)}{6}.</math> </br> Dotted portion of black line shows where <math>f(x)</math> is concave down. ]] In the diagram, point <math>p_1</math> on <math>f(x)</math> has coordinates <math>(x_1, f(x_1)).</math> Similarly, points <math>p_2, p_3</math> have coordinates <math>(x_2, f(x_2)),\ (x_3, f(x_3)).</math> <math>y'</math> has roots: <math>x_1 = -1;\ x_3 = -0.25.</math> Points <math>p_1, p_3</math> are stationary points. <math>y''</math> has roots: <math>x_1 = -1;\ x_2 = -0.5.</math> Points <math>p_1, p_2</math> are points of inflection. At point <math>p_3\ y''</math> is positive. <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is local minimum. {{RoundBoxTop|theme=2}} Summary: * Point <math>p_1</math> is both stationary point and point of inflection. * Point <math>p_2</math> is point of inflection. * Point <math>p_3</math> is both stationary point and local minimum. {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =The simplest quartic function= {{RoundBoxTop|theme=2}} [[File:0320quartic01.png|thumb|400px|'''Graph of simplest quartic function.''' </br> Point <math>p_1\ (-1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_2\ (1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_0\ (0, -1.1^4)</math> is <math>Y</math> intercept of <math>f(x).</math> ]] The simplest quartic function has coefficients <math>b = c = d = 0.</math> Red line in diagram has equation: <math>y = f(x) = x^4 - 1.1^4</math> First derivative (not shown) of <math>f(x):\ y' = g(x) = 4x^3.</math> When <math>x == 0,\ g(x) = 0.</math> There is a stationary point on <math>f(x)</math> when <math>x == 0,</math> point <math>p_0.</math> Second derivative (not shown) of <math>f(x):\ y'' = h(x) = 12x^2.</math> When <math>x == 0,\ h(x) = 0.</math> There is a point of inflection on <math>f(x)</math> when <math>x == 0.</math> For every non-zero value of <math>x,\ h(x)</math> is positive. To left and right of point <math>p_0,\ f(x)</math> is always concave up. Point <math>p_0</math> is both local minimum and absolute minimum. * Point <math>p_0</math> is stationary point and point of inflection and absolute minimum. Curve <math>f(x)</math> is useful for finding the fourth root of a real number. Solve: <math>x = N^{\frac{1}{4}}.</math> <math>x^4 = N.</math> <math>x^4 - N = 0.</math> This is equivalent to finding a root of function <math>y = j(x) = x^4 - N.</math> If you use Newton's method to find a root of <math>j(x),</math> this would be more efficient than solving <math>x = \sqrt{\sqrt{N}}.</math> {{RoundBoxBottom}} =Roots of equal absolute value= {{RoundBoxTop|theme=1}} The standard quartic function: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> For <math>x</math> in <math>(1)</math> substitute <math>(p+q).</math> Call this <math>(2).</math> For <math>x</math> in <math>(1)</math> substitute <math>(p-q).</math> Call this <math>(3).</math> Combine <math>(2)</math> and <math>(3)</math> to eliminate <math>q</math> and produce an equation in <math>p:</math> <math>(- 64aaa)pppppp+</math> <math>(- 96aab)ppppp+</math> <math>(- 32aac - 48abb)pppp+</math> <math>(- 32abc - 8bbb)ppp+</math> <math>(+ 16aae - 4abd - 4acc - 8bbc)pp+</math> <math>(+ 8abe - 2bbd - 2bcc)p+</math> <math>(+ add + bbe - bcd)\ =\ 0\ \dots\ (4).</math> We are interested in coefficient <math>0</math> of <math>(4):\ c_0 = add + bbe - bcd.</math> If <math>c_0 == 0,\ p=0</math> is a solution and function <math>(1)</math> has 2 roots of form <math>0 \pm q</math> where <math>q = \sqrt{\frac{-d}{b}}.</math> An example: {{RoundBoxTop|theme=3}} [[File:0320quartic02.png|thumb|400px|'''Graph of quartic function with 2 roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 12x^3 + 31x^2 + 48x -140}{45}.</math> <math>a,b,c,d,e = 1,-12,31,48,-140</math> <math>c_0 = add + bbe - bcd </math><math>= 1(48)(48) + (-12)(-12)(-140) - (-12)(31)(48) = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-48}{-12}} = \sqrt{4} = \pm 2.</math> The 2 roots of equal absolute value are: <math>2, -2.</math> {{RoundBoxBottom}} The method works with complex roots of equal absolute value: {{RoundBoxTop|theme=2}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with 2 complex roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}.</math> <math>a,b,c,d,e = 1,-3,-1,-27,-90</math> <math>c_0 = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-(-27)}{-3}} = \sqrt{-9} = \pm 3i.</math> The 2 roots of equal absolute value are: <math>3i, -3i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Equal roots= {{RoundBoxTop|theme=2}} Equal roots occur when the function and the slope of the function both equal zero. <math>ax^4 + bx^3 + cx^2 + dx + e = 0\ \dots\ (1)</math> <math>4ax^3 + 3bx^2 + 2cx + d = 0\ \dots\ (2)</math> Begin the process of reducing <math>(1),\ (2)</math> to linear functions. Combine <math>(1),\ (2)</math> to produce 2 cubic functions: <math>Fx^3 + Gx^2 + Hx + J\ \dots\ (1a)</math> where: <math>F = ad;\ G = bd - 4ae;\ H = cd - 3be;\ J = dd - 2ce.</math> <math>fx^3 + gx^2 + hx + j\ \dots\ (2a)</math> where: <math>f = 4a;\ g = 3b;\ h = 2c;\ j = d.</math> Combine <math>(1a),\ (2a)</math> to produce 2 quadratic functions: <math>Kx^2 + Lx + M\ \dots\ (1b)</math> where: <math>K = Gf -Fg;\ L = Hf -Fh;\ M = Jf - Fj.</math> <math>kx^2 + lx + m\ \dots\ (2b)</math> where: <math>k = Fj - Jf;\ l = Gj - Jg;\ m = Hj - Jh.</math> Combine <math>(1b),\ (2b)</math> to produce 2 linear functions: <math>Rx + S\ \dots\ (1c)</math> where: <math>R = Lk - Kl;\ S = Mk - Km.</math> <math>rx + s\ \dots\ (2c)</math> where: <math>r = Km-Mk;\ s = Lm - Ml.</math> From <math>(1c):\ x_1 = \frac{-S}{R}</math> From <math>(2c):\ x_2 = \frac{-s}{r}</math> If <math>x_1 == x_2:</math> <math>\frac{-S}{R} = \frac{-s}{r}</math> <math>Rs = rS</math> <math>Rs - Sr = 0.</math> The value <math>Rs - Sr</math> is in fact: <syntaxhighlight> + 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee - 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd + 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee + 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde + 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee - 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde - 3aabbccdddddd </syntaxhighlight> which, by removing values <math>aa, d</math><math>d</math> (common to all values), may be reduced to: <syntaxhighlight> status = ( + 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee - 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd + 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee + 384abccddde - 54abcddddd + 128acccccee - 80accccdde + 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee - 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde - 3bbccdddd ) </syntaxhighlight> If <math>status == 0,</math> there are at least 2 equal roots which may be calculated as shown below. {{RoundBoxTop|theme=2}} If coefficient <math>d</math> is non-zero, it is not necessary to calculate <math>status.</math> If coefficient <math>d == 0,</math> verify that <math>status = 0</math> before proceeding. {{RoundBoxBottom}} ==No equal roots== {{RoundBoxTop|theme=1}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with no equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}</math> <math>a,b,c,d,e = 1, -3, -1, -27, -90</math> <math>R, S = 15269148, -35977608</math> <math>x_1 = \frac{-S}{R} = \frac{35977608}{15269148} = 2.3562289133617\dots</math> <math>r, s = 35977608, -60634332</math> <math>x_2 = \frac{-s}{r} = \frac{60634332}{35977608} = 1.685335278543253\dots</math> <math>x_1 != x_2.</math> There are no equal roots. {{RoundBoxBottom}} ==Exactly 2 equal roots== {{RoundBoxTop|theme=1}} [[File:0329quartic04.png|thumb|400px|'''Graph of quartic function with exactly 2 equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 + 6x^3 - 48x^2 - 182x + 735}{100}</math> <math>a,b,c,d,e = 1, 6, -48, -182, 735</math> <math>R, S = -1027353600, -7191475200</math> <math>x_1 = \frac{-S}{R} = \frac{7191475200}{-1027353600} = -7</math> <math>r, s = 7191475200, 50340326400</math> <math>x_2 = \frac{-s}{r} = \frac{-50340326400}{7191475200} = -7</math> <math>x_1 = x_2 = -7.</math> There are 2 equal roots at <math>x = -7.</math> {{RoundBoxTop|theme=2}} The following 3 graphs show the steps that lead to calculation of equal roots at point <math>(-7, 0).</math> <gallery> File:0914quartic00.png|<small>Figure 1. graph of <math>f(x)</math> and 2 associated cubic functions.</small> File:0914quartic01.png|<small>Figure 2. graph of <math>f(x)</math> and 2 associated quadratic functions.</small> File:0914quartic02.png|<small>Figure 3. graph of <math>f(x)</math> and 2 associated linear functions.</small> </gallery> In all graphs, all curves have a common root at point <math>(-7, 0).</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=4}} See [[Quartic_function#Function_as_product_of_linear_function_and_cubic | "Function as product of linear function and cubic"]] above. To calculate all roots: <syntaxhighlight lang=python> # python code. a,b,c,d,e = 1,6,-48,-182,735 # The associated cubic: p = -7 A = a B = A*p + b C = B*p + c D = C*P + d # The associated quadratic: a1 = A b1 = a1*p + B c1 = b1*p + C a1,b1,c1 </syntaxhighlight> <syntaxhighlight> (1, -8, 15) </syntaxhighlight> Roots of quadratic function <math>g(x) = x^2 - 8x + 15</math> are <math>3, 5.</math> All roots of <math>f(x)</math> are <math>-7, -7, 3, 5.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Exactly 3 equal roots== {{RoundBoxTop|theme=1}} [[File:0320_2curves01.png|thumb|400px|'''Graph of quartic function with exactly 3 equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 2x^3 - 36x^2 + 162x - 189}{100}</math> <math>a,b,c,d,e = 1, -2, -36, 162, -189</math> <math>R, S = 0, 0\ \dots\dots\ (1c)</math> <math>r, s = 0, 0\ \dots\dots\ (2c)</math> In this case the calculation of <math>x_1, x_2</math> is not appropriate because there are more than 2 equal roots. Try equations <math>(1b), (2b).</math> Both of these are equivalent to: <math>y = g(x) = x^2 - 6x + 9,</math> blue line in diagram. Discriminant of <math>g(x) = (-6)^2 - 4(1)(9) = 0.\ g(x)</math> has two equal roots at <math>x = \frac{-(-6)}{2(1)} = 3.</math> Therefore <math>f(x)</math> has 3 equal roots at <math>x = 3.</math> {{RoundBoxBottom}} ==Four equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves01.png|thumb|400px|'''Graph of quartic function with 4 equal roots and corresponding cubic.''' </br> <math>g(x) = \frac{f'(x)}{4}.</math> ]] Red line in diagram is of function: <math>y = f(x) = x^4 - 20x^3 + 150x^2 - 500x + 625.</math> <math>a,b,c,d,e = 1, -20, 150, -500, 625</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> <math>K, L, M = 0, 0, 0</math> <math>k, l, m = 0, 0, 0</math> In this case <math>(1b), (2b), (1c), (2c)</math> are all null. This is the only case in which <math>(1b), (2b)</math> are null. <math>(1a), (2a)</math> are both equivalent to: <math>y = g(x) = x^3 - 15x^2 + 75x - 125,</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 4 equal roots at <math>x = 5.</math> {{RoundBoxBottom}} ==Two pairs of equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 6x^3 - 11x^2 + 60x + 100}{20}.</math> <math>a,b,c,d,e = 1, -6, -11, 60, 100</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> In this case <math>(1c), (2c)</math> are both null. <math>(1b), (2b)</math> are both equivalent to: <math>y = g(x) = \frac{x^2 - 3x - 10}{20},</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = -2</math> and one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 2 equal roots at <math>x = -2</math> and 2 equal roots at <math>x = 5.</math> {{RoundBoxTop|theme=2}} [[File:0408_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal, complex roots and corresponding quadratic.''' </br> <math>g(x) = (x-(3-2i))(x-(3+2i)) = x^2 - 6x + 13.</math> </br> <math>f(x) = (g(x))^2.</math> ]] This method is valid for complex roots. For example: <math>y = f(x) = x^4 - 12x^3 + 62x^2 - 156x + 169.</math> <math>a,b,c,d,e = 1,-12,62,-156,169.</math> In this case <math>(1c),\ (2c)</math> are both null. <math>(1b),\ (2b)</math> are both equivalent to: <math>y = g(x) = x^2 - 6x + 13,</math> blue line in diagram. Roots of <math>g(x)</math> are <math>3 \pm 2i.</math> <math>f(x)</math> has 2 roots equal to <math>3 + 2i</math> and 2 roots equal to <math>3 - 2i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Summmary== {{RoundBoxTop|theme=1}} {| class="wikitable" |- ! || No equal roots !! 2 equal roots !! 3 equal roots !! 4 equal roots !! 2 pairs of equal roots |- | Cubic: 1(a), 2(a) | different | different | different | same | different |- | Quadratic: 1(b), 2(b) | different | different | same, 1root | null | same, 2roots |- | Linear: 1(c), 2(c) | different | same | null | null | null |} {{RoundBoxBottom}} ==Caution== {{RoundBoxTop|theme=4}} [[File:0915quartic02.png|thumb|400px| Calculation of false equal roots.</br> In this example, method calculates 2 legitimate equal roots at <math>(4, 0)</math> and 2 false equal roots at <math>(7.543296089385474,0).</math> <math></math> ]] Black line in diagram has equation: <math>y = f(x) = 0.012684240362811794x^4</math> <math> -\ 0.19522392290249435x^3 </math> <math> +\ 0.7654478458049887x^2 + 0x - 3. </math> <math>f(x)</math> is a quartic function with exactly 2 equal roots and coefficient <math>d</math> missing. Calculation of equal roots of <math>f(x)</math> gives linear functions <math>(1c), (2c)</math> null and quadratic functions <math>(1b), (2b)</math> with equal roots of <math> (4,0), (7.543296089385474,0). </math> Usually, this indicates that <math>f(x)</math> should have 2 equal roots at <math>(4,0)</math> and 2 equal roots at <math>(7.543296089385474,0).</math> It is obvious that <math>7.543296089385474</math> is not a root of <math>f(x).</math> When <math>x = 7.543296089385474,</math> slope of derivative <math>g(x) = 0.</math> Value of <math>f(x)\ != 0.</math> This example indicates that it would be wise to verify that calculated equal roots are in fact valid roots of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Depressed quartic= {{RoundBoxTop|theme=1}} A depressed quartic is any quartic function with any one or more of coefficients <math>b,c,d</math> missing. Within this section a depressed quartic has coefficient <math>b</math> missing. To produce the depressed quartic: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> <math>y = \frac{(4^4 a^3)(ax^4 + bx^3 + cx^2 + dx + e)}{4^4 a^3}\ \dots\ (2)</math> Let <math>x = \frac{-b + t}{4a}.</math> Substitute in <math>(2),</math> expand and simplify: <math>y = \frac{t^4 + At^2 + Bt + C}{4^4a^3}\ \dots\ (3)</math> where: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> When equated to <math>0,\ (3)</math> becomes the depressed equation: <math>t^4 + At^2 + Bt + C = 0\ \dots\ (4).</math> Be prepared for the possibility that any 1 or more of <math>A, B, C</math> may be zero. ==Coefficient B missing== {{RoundBoxTop|theme=2}} [[File:0322_3curves01.png|thumb|400px|'''Graph of quartic function that resembles a quadratic.''' ]] If coefficient <math>B == 0,\ (4)</math> becomes a quadratic in <math>t^2:</math> <math>t^4 + At^2 + C = 0.</math> <math>(1)</math> has the appearance of a quadratic. The black line: <math>y = f(x) = \frac{x^4 - 4x^3 + 9x^2 - 10x + 5}{10}</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>= 8( 8(-10) - 4(-4)(9) + -64 )</math> <math>= 8(-80 + 144 - 64)</math> <math>= 8(0) = 0.</math> The red line: <math>y' = g(x) = \frac{4x^3 - 12x^2 + 18x - 10}{10}</math> <math>y' = g(x) = 0</math> where <math>x = 1.</math> The grey line: <math>y'' = h(x) = \frac{12x^2 - 24x + 18}{10}</math> * Absolute minima of <math>f(x)</math> and of <math>h(x)</math> and point of inflection of <math>g(x)</math> occur where <math>x = \frac{-b}{4} = 1.</math> * <math>y''</math> is always positive. <math>f(x)</math> is always concave up. * <math>f(1 + p) = f(1-p).</math> {{RoundBoxTop|theme=3}} If <math>(1)</math> contains 2 pairs of equal roots, coefficient <math>B = 0.</math> The converse is not necessarily true. If <math>(1)</math> contains 4 equal roots, coefficients <math>A = B = C = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Coefficient C missing== {{RoundBoxTop|theme=2}} [[File:0501quartic01.png|thumb|400px|'''Graph of quartic function with coefficient C of depressed function missing.''' </br> <math>Y</math> axis compressed for clarity. ]] If coefficient <math>C == 0,\ (4)</math> becomes: <math>t^4 + At^2 + Bt = t(t^3 + At + B) = 0</math> in which case <math>t = 0</math> is a solution and <math>x = \frac{-b}{4a}</math> is a root of <math>(1).</math> Curve (red line) in example has equation: <math>y = f(x) = 8x^4 + 16x^3 + 24x^2 + 89x + 40.</math> Coefficients of depressed function are: <syntaxhighlight lang=python> # python code a,b,c,d,e = 8, 16, 24, 89, 40 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b A,B,C </syntaxhighlight> <syntaxhighlight> (1536, 299008, 0) </syntaxhighlight> Coefficient <math>C</math> of depressed function is missing. <math>t = 0</math> is a solution. Using <math>x = \frac{-b + t}{4a},</math> one root of <math>f(x) = \frac{-16 + 0}{4(8)} = -0.5.</math> {{RoundBoxBottom}} ==Resolvent cubic== {{RoundBoxTop|theme=1}} This section introduces a special cubic function called "resolvent" because it helps to resolve a requirement, the calculation of the roots of the quartic. The depressed quartic: <math>t^4 + At^2 + Bt + C\ \dots\ (1)</math> For <math>t</math> substitute <math>(u+v)\ \dots\ (2)</math> For <math>t</math> substitute <math>(u-v)\ \dots\ (3)</math> <math>(2)+(3):\ 2Auu + 2Avv + 2Bu + 2C + 2uuuu + 12uuvv + 2vvvv\ \dots\ (4)</math> Simplify <math>(4):\ Auu + AV + Bu + C + uuuu + 6uuV + VV\ \dots\ (4a)</math> <math>(2)-(3):\ 4Auv + 2Bv + 8uuuv + 8uvvv\ \dots\ (5)</math> Simplify <math>(5):\ 2Au + B + 4uuu + 4uV\ \dots\ (5a)</math> From <math>(5a):\ 4uV = -(2Au + B + 4uuu)\ \dots\ (5b)</math> <math>(4a)*4u4u:\ 4u4uAuu + A4u(4uV) + 4u4uBu + 4u4uC + 4u4uuuuu + 6uu4u(4uV) + (4uV)(4uV)\ \dots\ (6)</math> In <math>(6)</math> replace <math>4uV</math> with <math>(-(2Au + B + 4uuu)),</math> expand, simplify, gather like terms and result is: <math>Pu^6 + Qu^4 + Ru^2 + S</math> or {{RoundBoxTop|theme=2}} <math>PU^3 + QU^2 + RU + S\ \dots\ (7)</math> where: <math>U = u^2</math> <math>P = 64</math> <math>Q = 32A</math> <math>R = 4A^2 - 16C</math> <math>S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} From <math>(5b):\ V = v^2 = \frac{-(2Au + B + 4uuu)}{4u} = -(\frac{A}{2} + U) - \frac{B}{4u}\ \dots\ (8)</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Some simple changes reduce the number of calculations and also the sizes of coefficients <math>P, Q, R, S.</math> <math>A2 = 8ac - 3b^2</math> where <math>A2 = \frac{A}{2}</math> <math>B4 = 16a^2 d - 8abc + 2b^3</math> where <math>B4 = \frac{B}{4}</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> Then: <math>P = 64</math> <math>Q = 32(A2)(2) = 64A2</math> <math>R = 4(A2)(2)(A2)(2) - 16C = 16A2^2 - 16C</math> <math>S = -(B4)(4)(B4)(4) = -16 B4^2</math> Divide all 4 coefficients by <math>16:</math> <math>P = 4</math> <math>Q = 4A2</math> <math>R = A2^2 - C</math> <math>S = -B4^2</math> <math>V = v^2 = -(A2 + U) - \frac{B4}{u}.</math> A close examination of coefficients <math>R, S</math> shows that both coefficients are always exactly divisible by <math>4.</math> Therefore, all four coefficients may be defined as follows: <math>P = 1</math> <math>Q = A2</math> <math>R = \frac{A2^2 - C}{4}</math> <math>S = \frac{-B4^2}{4}</math> Actual calculations of values <math>R, S</math> are: <syntaxhighlight lang=python> # python code. R = ( (16)*(a)*(a)*(b)*(d) + (16)*(a)*(a)*(c)*(c) + (3)*(b)*(b)*(b)*(b) + (-16)*(a)*(b)*(b)*(c) + (-64)*(a)*(a)*(a)*(e) ) S = ( (64)*(a)*(a)*(a)*(b)*(c)*(d) + (8)*(a)*(b)*(b)*(b)*(b)*(c) + (-16)*(a)*(a)*(b)*(b)*(b)*(d) + (-16)*(a)*(a)*(b)*(b)*(c)*(c) + (-1)*(b)*(b)*(b)*(b)*(b)*(b) + (-64)*(a)*(a)*(a)*(a)*(d)*(d) ) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =Solving quartic equation= This section presents 4 examples that show how to use the depressed quartic and the resolvent cubic to solve the quartic equation. ==Four real roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves01.png|thumb|400px|'''Graphs of quartic function with 4 real roots and associated resolvent cubic.''' </br>Resolvent cubic contains 3 real, positive roots. ]] Calculate roots of: <math>y = f(x) = x^4 - x^3 - 19x^2 - 11x + 30</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,-1,-19,-11,30 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -310 -1320 6669 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 -9920 277696 -1742400 </syntaxhighlight> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = 64x^3 - 9920x^2 + 277696x - 1742400.</math> There are 3 real, positive roots: <math>9, 25, 121.</math> Using 3 roots of <math>g(x),</math> calculate 4 roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code for U in (9, 25, 121) : print ('\nU =', U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** .5 for t in (u+v, u-v) : x = (-b+t) / (4*a) y = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x:',x, '; y:',y) </syntaxhighlight> <syntaxhighlight> U = 9 x: 5.0 ; y: 0.0 x: -3.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 U = 25 x: 5.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -3.0 ; y: 0.0 U = 121 x: 5.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: -3.0 ; y: 0.0 </syntaxhighlight> Roots of <math>f(x)</math> are: <math>5, 1, -2, -3.</math> {{RoundBoxBottom}} All 3 values of <math>U</math> produce the same results, but not in same sequence. It is not necessary to calculate all 3 roots of resolvent cubic. Any one non-zero root is sufficient to do the job. ==Two real and two complex roots== ===Example 1=== {{RoundBoxTop|theme=4}} [[File:0323_2curves02.png|thumb|400px|'''Graphs of quartic function with 2 real and 2 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 + 2x^3 + 18x^2 - 70x - 87</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,2,18,-70,-87 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> 264 -5568 -12208 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 8448 474112 -31002624 </syntaxhighlight> {{RoundBoxBottom}} Calculate one real root of cubic function: <math>y = g(x) = 64x^3 + 8448x^2 + 474112x -31002624.</math> <math>36</math> is one real root. Choose <math>U = 36.</math> Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 36 u1 = U**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) if V >= 0 : v = V**.5 else : v = 1j * (-V)**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =',x, '; sum =',sum ) </syntaxhighlight> <syntaxhighlight> x = 3.0 ; sum = 0.0 x = -1.0 ; sum = 0.0 x = (-2+5j) ; sum = 0j x = (-2-5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=4}} [[File:0321_2curves00.png|thumb|400px| Coefficient <math>d</math> of <math>g(x) = 0.</math> <math></math> ]] Calculate roots of: <math>y = f(x)</math> <math>= 3x^4 - 6x^3 - 41x^2 +44x - 189</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 3, -6, -41, 44, -189 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -2184, 0, -1229040 </syntaxhighlight> Notice that coefficient <math>B = 0.</math> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 1, -1092, 605376, 0 </syntaxhighlight> Notice that coefficient <math>S = 0.</math> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = x^3 - 1092x^2 + 605376x + 0.</math> Roots are <math>0, 546 \pm 554.3103823671355j.</math> Value <math>0</math> cannot be used because it will cause error <code>Divide by zero</code> at statement <code>V = -(A/2 + U) - B/(4*u)</code>. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 546+554.3103823671355j print ('\nU =',U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** 0.5 s1 = '\nu,v' print (s1,eval(s1)) for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> U = (546+554.3103823671355j) u,v ((25.729935131257832+10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (4.788322521876306 + 0j) ; sum = (1.9895196601282805e-13 + 0j) x = (0.5 + 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) u,v ((-25.729935131257832-10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (0.5 - 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) x = (-3.7883225218763052 + 0j) ; sum = (1.7053025658242404e-13 + 0j) </syntaxhighlight> Values of <math>x</math> are: <math>-3.7883225218763052, 4.788322521876306, 0.5 \pm 1.795283650280614j</math> <math></math><math></math> {{RoundBoxBottom}} ====Depressed quartic as quadratic==== In this example coefficient <math>B</math> of depressed quartic <math>= 0.</math> Therefore, resolvent cubic can be ignored and depressed quartic processed as quadratic in <math>T = t^2.</math> <math>t^4 - 2184t^2 + (0)t - 1229040</math> <math>T^2 - 2184T - 1229040</math> where <math>T = t^2.</math> Solutions of this quadratic are: <math>T_1, T_2 = 2648.1182474349434, -464.11824743494344</math> <syntaxhighlight lang=python> T1,T2 = 2648.1182474349434, -464.11824743494344 t1 = T1 ** 0.5; t2 = ((-T2) ** 0.5) * 1j for t in (t1,-t1,t2,-t2) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> x = 4.788322521876305 ; sum = -1.7053025658242404e-13 x = -3.788322521876305 ; sum = -1.7053025658242404e-13 x = (0.5 + 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) x = (0.5 - 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) </syntaxhighlight> or <syntaxhighlight> x = 0.5 ± 4.288322521876305, 0.5 ± 1.7952836502806138j </syntaxhighlight> With precision of 15, values of <math>x</math> are same as those shown above. When roots of quartic function are of form <code>p ± q, p ± r,</code> coefficient <math>B</math> of depressed function <math>= 0.</math> ==Four complex roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves03.png|thumb|400px|'''Graphs of quartic function with 4 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 - 20x^3 + 408x^2 + 2296x + 18020</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 4128 344064 9683200 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 64 132096 -86769664 -118380036096 </syntaxhighlight> {{RoundBoxBottom}} Calculate one root of cubic function: <math>y = g(x) = 64x^3 + 132096x^2 - 86769664x - 118380036096.</math> There are 3 real roots: <math>-2304, -784, 1024.</math> Choose <math>U = -784.</math> Negative <math>U</math> is chosen here to show that any 1 of the roots produces the correct result. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = -784 u1 = 1j * (-U)**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) v = V**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> # python expresses complex numbers with 'j'. x = (13+19j) ; sum = 0j x = (-3-5j) ; sum = 0j x = (13-19j) ; sum = 0j x = (-3+5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} =Quartic formula= The substitutions made above can be used to produce a formula for the solution of the quartic equation. See main articles [https://en.wikipedia.org/wiki/Quartic_equation#The_general_case "The general case" ] or [https://en.wikipedia.org/wiki/Quartic_function#General_formula_for_roots "General formula for roots."] {{RoundBoxTop|theme=2}} Both links above point to formula for equation <math>x^4 + ax^3 + bx^2 + cx + d = 0.</math> {{RoundBoxBottom}} Given quartic equation: <math>ax^4 + bx^3 + cx^2 + dx + e = 0,</math> calculate the 4 values of <math>x.</math> <math>x = \frac{-b + t}{4a}</math> where: {{RoundBoxTop|theme=2}} Coefficients of depressed quartic: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of resolvent cubic: <math>a_1 = P = 64</math> <math>b_1 = Q = 32A</math> <math>c_1 = R = 4A^2 - 16C</math> <math>d_1 = S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of depressed cubic: <math>A_1 = 9a_1 c_1 - 3b_1^2</math> <math>B_1 = 27a_1^2d_1 - 9a_1b_1c_1 + 2b_1^3</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of resolvent cubic: <math>C_1 = \frac{-A_1}{3} = b_1^2 - 3a_1 c_1</math> <math>\Delta = B_1^2 - 4C_1^3\ \dots\dots\ \Delta</math> may be negative. <math>\delta = \sqrt{\Delta}</math> <math>W = \frac{-B_1 + \delta}{2}</math> <math>w = \sqrt[3]{W}</math> <math>t_1 = w + \frac{C_1}{w}</math> <math>U = \frac{-b_1 + t_1}{3a_1}</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of quartic: <math>u = \sqrt{U}\ \dots\dots\ u</math> may be positive or negative. <math>V = -(\frac{A}{2} + U) - \frac{B}{4u}</math> <math>v = \sqrt{V}\ \dots\dots\ v</math> may be positive or negative. <math>t = u + v</math> {{RoundBoxBottom}} Formula above produces one value of <math>x.</math> Python code below utilizes <math>\pm \sqrt{U}</math> and <math>\pm \sqrt{V}</math> to produce 4 values of <math>t</math> and then, four values of <math>x.</math> ==An example:== {{RoundBoxTop|theme=4}} [[File:0330quartic01.png|thumb|400px|'''Graph of quartic function with 2 real roots.'''</br><math>Y</math> axis compressed for clarity.]] Calculate roots of <math>f(x) = 4x^4 + 4x^3 - 75x^2 - 776x - 1869.</math> <syntaxhighlight lang=python> # Python code. a,b,c,d,e = 4, 4, -75, -776, -1869 values_of_t = [ t # Coefficients of depressed quartic: for A in (16*a*c - 6*b*b,) for B in (64*a*a*d -32*a*b*c + 8*b*b*b,) for C in (256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b,) # Coefficients of resolvent cubic: for a1 in (64,) for b1 in (32*A,) for c1 in (4*A*A - 16*C,) for d1 in (-B*B,) for U in [ # The resolvent cubic: (-b1+t1)/(3*a1) # Coefficients of depressed resolvent cubic: for A1 in (9*a1*c1 - 3*b1*b1,) for B1 in (27*a1*a1*d1 - 9*a1*b1*c1 + 2*b1*b1*b1,) # One root of resolvent cubic: for C1 in (-A1/3,) for Δ in (B1*B1 - 4*C1*C1*C1,) for δ in (Δ**0.5,) for W in ((-B1 + δ)/2,) for w in (W**(1/3),) for t1 in (w + C1/w,) # See note below. ] # Prepare to calculate 4 values of t. for u1 in (U**.5,) for v1 in ( -(A/2 + U) ,) # Calculate 4 values of t. for u in (u1, -u1,) for V in ( v1 - B/(4*u),) for v in (V**.5,) for t in (u+v, u-v) ] print ('values_of_t =', values_of_t) </syntaxhighlight> {{RoundBoxTop|theme=4}} <syntaxhighlight> values_of_t = [116, -44, (-36+64j), (-36-64j)] </syntaxhighlight> Because <math>f(t)</math> is a depressed quartic function, sum of four <code>values_of_t</code> <math>= 116 - 44 - 36(2) = 0.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> # Python code. # Calculate 4 separate roots. values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] print ('values_of_x =', values_of_x) </syntaxhighlight> <syntaxhighlight lang=python> values_of_x = [7, -3, (-2.5 + 4j), (-2.5 - 4j)] </syntaxhighlight> In python the imaginary part of a complex number is shown with <math>j</math> instead of <math>i.</math> If <math>A == B == C == 0,\ f(x)</math> contains 4 equal roots and <math>x = \frac{-b}{4a}.</math> If <math>f(x)</math> contains 3 or more equal roots, statement <code>for t1 in (w + C1/w,)</code> fails with divisor <math>w = 0.</math> Before using this formula, check for equal roots as in [[Quartic_function#Exactly_3_equal_roots | "Exactly 3 equal roots"]] above. {{RoundBoxTop|theme=1}} Values displayed above have been edited slightly. Actual calculated values were: <syntaxhighlight> values_of_x = [7.000000000000001, -3.0000000000000044, (-2.499999999999998+4.000000000000001j), (-2.4999999999999987-4.000000000000001j)] </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ==In practice== {{RoundBoxTop|theme=5}} The following Python code implements the quartic formula. However, under statement <code>if B4 == 0 :</code> there is code that processes the depressed quartic as a quadratic in <math>T = t^2.</math> This ensures that execution of formula does not fail with error <code>Divide by zero</code> at statement <code>for V in ( v1 - B4/u,).</code> {{RoundBoxTop|theme=4}} <syntaxhighlight lang=python> # python code import cmath cxSqrt = cmath.sqrt # Square root of complex number. def rootsOfQuartic (abcde) : ''' x1,x2,x3,x4 = rootsOfQuartic ((a,b,c,d,e)) Each member of input must be int or float or Decimal object. Int or Decimal object in input is quietly converted to float. Output may be None. ''' def formatResults (x1x2x3x4) : ''' This function improves appearance of results. (8 + 0j) becomes 8.0 ''' values_of_x = list (x1x2x3x4) for p in (0,1,2,3) : v = values_of_x[p] if isinstance (v, complex) and (v.imag == 0) : values_of_x[p] = v.real return values_of_x status = 0 try : a,b,c,d,e = [ float(v) for v in abcde ] except : status = 1 if status : print ('rootsOfQuartic () 1: Error creating coefficients a,b,c,d,e.') return None if a == 0 : print ('rootsOfQuartic () 2: Coefficient a must be non-zero.') return None # Coefficients of depressed quartic, modified. # A = 16*a*c - 6*b*b A2 = 8*a*c - 3*b*b # B = 64*a*a*d - 32*a*b*c + 8*b*b*b B4 = 16*a*a*d - 8*a*b*c + 2*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b if B4 == 0 : # B = 0. # Result returned from this section is type tuple, indicating that coefficient B4 = 0. if A2==C==0 : # 4 equal roots. root = -b/(4*a) return tuple(formatResults((root,root,root,root))) # t**4 + At**2 + (0)t + C # Depressed quartic is quadratic in T: # T**2 + AT + C where T = t**2 # T**2 + 2(A2)T + C where A = 2(A2) # -2(A2) +/- (4(A2)(A2) - 4C)**0.5 # T = -------------------------------- = -A2 +/- ((A2)(A2) - C)**0.5 # 2 disc = A2*A2 - C if disc >= 0 : root = disc ** 0.5 else : root = ((-disc) ** 0.5) * 1j T1 = -A2 - root ; T2 = -A2 + root t1 = cxSqrt(T1) ; t2 = cxSqrt(T2) values_of_t = (t1,-t1,t2,-t2) values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] return tuple(formatResults(values_of_x)) # B4 is non-zero. Therefore, all of (S, U, u) are non-zero. P,Q,R,S = 1, A2, (A2*A2 - C)/4, -B4*B4/4 # str1 = 'P,Q,R,S' ; print (str1, eval(str1)) U = oneRootOfCubic((P,Q,R,S)) # Resolvent cubic. if U > 0 : sqrtU = U ** 0.5 elif U == 0 : # This should not happen. print ('rootsOfQuartic () 3: Internal error.') return None else : sqrtU = ((-U) ** 0.5) * 1j v1 = -(A2+U) values_of_t = [ t for u in (sqrtU, -sqrtU) for V in ( v1 - B4/u, ) for v in ( cxSqrt(V), ) for t in (u+v, u-v) ] values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] # Result returned from this section is type list, indicating that coefficient B4 != 0. return formatResults(values_of_x) </syntaxhighlight> For function <code>oneRootOfCubic()</code> see [[Cubic_function#In_practice | Cubic_function: In_practice.]] {{RoundBoxBottom}} ===Examples=== {{RoundBoxTop|theme=2}} Python function <code>equalRoots()</code> below implements <code>status</code> as presented under [https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above. <syntaxhighlight lang=python> # python code def equalRoots(abcde) : ''' This function returns True if quartic function contains at least 2 equal roots. ''' a,b,c,d,e = abcde aa = a*a ; aaa = aa*a bb = b*b ; bbb = bb*b ; bbbb = bb*bb cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd ee = e*e ; eee = ee*e ; eeee = ee*ee v1 = ( +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd +20*bb*ccc*dd*e ) v2 = ( -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd ) return (v1+v2) == 0 t1 = ( ((1, -1, -19, -11, 30), '4 unique, real roots.'), ((4, 4,-119, -60, 675), '4 unique, real roots, B4 = 0.'), ((1, 6, -48,-182, 735), '2 equal roots.'), ((1,-12, 50, -84, 45), '2 equal roots. B4 = 0.'), ((1,-20, 146,-476, 637), '2 equal roots, 2 complex roots.'), ((1,-12, 58,-132, 117), '2 equal roots, 2 complex roots. B4 = 0.'), ((1, -2, -36, 162, -189), '3 equal roots.'), ((1,-20, 150,-500, 625), '4 equal roots. B4 = 0.'), ((1, -6, -11, 60, 100), '2 pairs of equal roots, B4 = 0.'), ((4, 4, -75,-776,-1869), '2 complex roots.'), ((1,-12, 33, 18, -208), '2 complex roots, B4 = 0.'), ((1,-20, 408,2296,18020), '4 complex roots.'), ((1,-12, 83, -282, 442), '4 complex roots, B4 = 0.'), ((1,-12, 62,-156, 169), '2 pairs of equal complex roots, B4 = 0.'), ) for (abcde, comment) in t1 : print () fourRoots = rootsOfQuartic (abcde) print (comment) print (' Coefficients =', abcde) print (' Four roots =', fourRoots) print (' Equal roots detected:', equalRoots(abcde)) # Check results. a,b,c,d,e = abcde for x in fourRoots : # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0 sum = (a*x**4 + b*x**3 + c*x**2 + d*x + e) if sum : # Create exception. 1/0 </syntaxhighlight> <syntaxhighlight> 4 unique, real roots. Coefficients = (1, -1, -19, -11, 30) Four roots = [5.0, 1.0, -2.0, -3.0] Equal roots detected: False 4 unique, real roots, B4 = 0. Coefficients = (4, 4, -119, -60, 675) Four roots = (2.5, -3.0, 4.5, -5.0) Equal roots detected: False 2 equal roots. Coefficients = (1, 6, -48, -182, 735) Four roots = [5.0, 3.0, -7.0, -7.0] Equal roots detected: True 2 equal roots. B4 = 0. Coefficients = (1, -12, 50, -84, 45) Four roots = (3.0, 3.0, 5.0, 1.0) Equal roots detected: True 2 equal roots, 2 complex roots. Coefficients = (1, -20, 146, -476, 637) Four roots = [7.0, 7.0, (3+2j), (3-2j)] Equal roots detected: True 2 equal roots, 2 complex roots. B4 = 0. Coefficients = (1, -12, 58, -132, 117) Four roots = ((3+2j), (3-2j), 3.0, 3.0) Equal roots detected: True 3 equal roots. Coefficients = (1, -2, -36, 162, -189) Four roots = [3.0, 3.0, 3.0, -7.0] Equal roots detected: True 4 equal roots. B4 = 0. Coefficients = (1, -20, 150, -500, 625) Four roots = (5.0, 5.0, 5.0, 5.0) Equal roots detected: True 2 pairs of equal roots, B4 = 0. Coefficients = (1, -6, -11, 60, 100) Four roots = (5.0, -2.0, 5.0, -2.0) Equal roots detected: True 2 complex roots. Coefficients = (4, 4, -75, -776, -1869) Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)] Equal roots detected: False 2 complex roots, B4 = 0. Coefficients = (1, -12, 33, 18, -208) Four roots = ((3+2j), (3-2j), 8.0, -2.0) Equal roots detected: False 4 complex roots. Coefficients = (1, -20, 408, 2296, 18020) Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)] Equal roots detected: False 4 complex roots, B4 = 0. Coefficients = (1, -12, 83, -282, 442) Four roots = ((3+5j), (3-5j), (3+2j), (3-2j)) Equal roots detected: False 2 pairs of equal complex roots, B4 = 0. Coefficients = (1, -12, 62, -156, 169) Four roots = ((3+2j), (3-2j), (3+2j), (3-2j)) Equal roots detected: True </syntaxhighlight> When description contains note <math>B4 = 0,</math> depressed quartic was processed as quadratic in <math>t^2.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Two Conic Sections= Examples of conic sections include: ellipse, circle, parabola and hyperbola. This section presents examples of two conic sections, circle and ellipse, and how to calculate the coordinates of the point/s of intersection, if any, of the two sections. Let one section with name <math>ABCDEF</math> have equation <math>Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.</math> Let other section with name <math>abcdef</math> have equation <math>ax^2 + by^2 + cxy + dx + ey + f = 0.</math> Because there can be as many as 4 points of intersection, a special "resolvent" quartic function is used to calculate the <math>x</math> coordinates of the point/s of intersection. Coefficients of associated "resolvent" quartic are calculated as follows: <syntaxhighlight lang=python> # python code def intersection_of_2_conic_sections (abcdef, ABCDEF) : ''' A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF) where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function: y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_ ''' A,B,C,D,E,F = ABCDEF a,b,c,d,e,f = abcdef G = ((-1)*(B)*(a) + (1)*(A)*(b)) H = ((-1)*(B)*(d) + (1)*(D)*(b)) I = ((-1)*(B)*(f) + (1)*(F)*(b)) J = ((-1)*(C)*(a) + (1)*(A)*(c)) K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c)) L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c)) M = ((-1)*(E)*(f) + (1)*(F)*(e)) g = ((-1)*(C)*(b) + (1)*(B)*(c)) h = ((-1)*(E)*(b) + (1)*(B)*(e)) A_ = (J)*(g) + (G)*(G) B_ = (J)*(h) + (K)*(g) + (2)*(G)*(H) C_ = (K)*(h) + (L)*(g) + (2)*(G)*(I) + (H)*(H) D_ = (L)*(h) + (M)*(g) + (2)*(H)*(I) E_ = (M)*(h) + (I)*(I) str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_) print (str1) return A_,B_,C_,D_,E_ </syntaxhighlight> == With no common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections06.png|thumb|400px|'''Two conic sections with no common point.''' </br> Resolvent quartic function (black curve) has no real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 80 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.98x^2 + 6850.532x - 22836.7009.</math> <math>f(x)</math> has no real roots. Therefore, there is no point of intersection. {{RoundBoxBottom}} ==With one common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections05.png|thumb|400px|'''Two conic sections with one common point.''' </br> Resolvent quartic function (black curve) has two equal, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 73 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.84x^2 + 7456.48x - 24355.36.</math> Roots of <math>f(x)</math> are: <math>(-14-22.978250586152114j),</math> <math>(-14+22.978250586152114j),</math> <math>5.8, 5.8.</math> <math>f(x)</math> has 2 equal, real roots at <math>x = 5.8,</math> effectively 1 real root where <math>x = 5.8</math> Therefore, there is one point of intersection where <math>x = 5.8.</math> {{RoundBoxBottom}} ==With two common points== ===Example 1=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections04.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two unique, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 64 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.66x^2 + 8235.556x - 26681.1841.</math> Roots of <math>f(x)</math> are: <math>(-14.361578825892241-23.341853011785357j),</math> <math>(-14.361578825892241+23.341853011785357j),</math> <math>4.59885619413921, 7.72430145764527.</math> <math>f(x)</math> has 2 unique, real roots at <math>x = 4.59885619413921, 7.72430145764527.</math> Therefore, there are two points of intersections where <math>x = 4.59885619413921, 7.72430145764527.</math> {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections01.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two pairs of equal roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 18.8x - 1.6y + 53 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 + 37.6x^3 - 504.24x^2 + 2835.04x - 5685.16.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 13, 13.</math> <math>f(x)</math> has 2 pairs of equal roots at <math>x = 5.8, 13,</math> effectively 2 real roots. Therefore, there are two points of intersection where <math>x = 5.8, 13.</math> {{RoundBoxBottom}} ==With 3 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections03.png|thumb|400px|'''Two conic sections with three common points.''' </br> Resolvent quartic function (black curve) has one pair of equal roots and 2 unique, real roots. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 17.6x - 3.2y + 55 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 6.83589838486224, 13.7641016151377 .</math> <math>f(x)</math> has 1 pair of equal roots at <math>x = 5.8</math> and 2 unique, real roots at <math>x = 6.83589838486224, 13.7641016151377,</math> effectively 3 real roots. Therefore, there are three points of intersection where <math>x = 5.8, 6.83589838486224, 13.7641016151377.</math> {{RoundBoxBottom}} ==With 4 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections02.png|thumb|400px|'''Two conic sections with four common points.''' </br> Resolvent quartic function (black curve) has 4 unique, real roots. </br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 18.8x - 1.6y + 62.99 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x)</math><math>= -x^4 + 37.6x^3 - 504.4398x^2 + 2838.79624x - 5544.61147921.</math> Roots of <math>f(x)</math> are:<math>4.36661032156283,</math><math>8.77936456353008,</math> <math>10.0206354364699,</math><math>14.4333896784371.</math> <math>f(x)</math> has 4 real roots as shown above. Therefore, there are four points of intersection where <math>x = 4.36661032156283,</math> <math>8.77936456353008,</math><math>10.0206354364699,</math><math>14.4333896784371.</math> {{RoundBoxBottom}} ===Values of y=== Section above calculates <math>x</math> coordinates of the four common points. This section calculates <math>y</math> coordinates. Here is the lazy way to do it: {{RoundBoxTop|theme=2}} [[File:0818ResQuartic.png|thumb|400px|'''Graphs of 2 curves, red and blue, with coefficients of <math>x,y</math> reversed.''' </br> Roots of "resolvent" quartic, black curve, are interpreted as values of <math>y.</math> ]] <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 # Coefficients of x,y are reversed. abcde = intersection_of_2_conic_sections ((b,a,0,e,d,f), (B,A,C,E,D,F)) values_of_y = rootsOfQuartic (abcde) print (values_of_y) </syntaxhighlight> <syntaxhighlight> [1.6215767432213488, -0.021576743221348723, 5.862095579401657, -4.262095579401658] </syntaxhighlight> By inspection of the diagram above, it is obvious that: <syntaxhighlight lang=python> # python code four_common_points = ( (4.36661032156283, 1.6215767432213488), (8.77936456353009, -4.262095579401658), (10.0206354364699, 5.862095579401657), (14.4333896784372, -0.021576743221348723), ) </syntaxhighlight> Check results: <syntaxhighlight lang=python> # python code def sum_zero (input) : ''' This function calculates sum of all values in input. If sum is very close to 0, this function returns 0. ''' sum = 0 for v in input : sum += v if sum == 0 : return sum tolerance = 1e-14 max = sorted([ abs(v) for v in input ])[-1] if abs(sum) < tolerance*max : return 0 return sum # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 for (x,y) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum1 = sum_zero(values) if sum1 : print ('Error: sum1 =',sum1) values = a*x**2, b*y**2, d*x, e*y, f sum2 = sum_zero(values) if sum2 : print ('Error: sum2 =',sum2) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. Check results with coefficients of <math>x, y</math> reversed: <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = B,A,C,E,D,F # The circle: a,b,c,d,e,f = b,a,c,e,d,f for (y,x) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum3 = sum_zero(values) if sum3 : print ('Error: sum3 =',sum3) values = a*x**2, b*y**2, d*x, e*y, f sum4 = sum_zero(values) if sum4 : print ('Error: sum4 =',sum4) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. {{RoundBoxBottom}} =Links to related topics= {{RoundBoxTop|theme=1}} [[Cubic_function#Cubic_formula | "Cubic formula"]] [[Complex_square_root | "Complex square root" ]] {{RoundBoxBottom}} [[Category:Mathematics]] 2weyrc3k9j2subie343touuwp6f9076 2844181 2844180 2026-11-03T11:28:39Z ThaniosAkro 2805358 /* With 3 common points */ 2844181 wikitext text/x-wiki The quartic function is the bridge between the cubic function and more advanced functions such as the quintic and sextic. =Objective= {{RoundBoxTop|theme=3}} [[file:Books-aj.svg_aj_ashton_01f.png|right|100px]] * Present quartic function and quartic equation. * Introduce the concept of roots of equal absolute value. * Show how to predict and calculate equal roots, techniques that will be useful when applied to higher order functions. * Simplify the depressed quartic. * Show that the quartic equation is effectively solved when at least one root is known. * Present the "resolvent" cubic function. * Show how to derive and use the quartic formula. {{RoundBoxBottom}} =Lesson= {{RoundBoxTop|theme=2}} ==Introduction== The quartic function is the sum of powers of <math>x</math> from <math>0</math> through <math>4</math>: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx^1 + ex^0</math> usually written as: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx + e.</math> If <math>e == 0</math> the function becomes <math>x(ax^3 + bx^2 + cx + d).</math> Within this page we'll say that: * both coefficients <math>a, e</math> must be non-zero, * coefficient <math>a</math> must be positive (simply for our convenience), * all coefficients must be real numbers, accepting that the function may contain complex roots. The quartic equation is the quartic function equated to zero: <math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>. Roots of the function are values of <math>x</math> that satisfy the quartic equation. {{RoundBoxTop|theme=1}} Because the function is "quartic" (maximum power of <math>x</math> is <math>4</math>), the function contains exactly <math>4</math> roots, an even number of complex roots and an even number of real roots. Other combinations of real and complex roots are possible, but they produce complex coefficients. {{RoundBoxBottom}} {{RoundBoxTop|theme=7}} [[File:0312quartic01a.png|thumb|400px|'''Graph of typical quartic function showing minima and maximum.''']] The figure shows a typical quartic function. The function crosses the <math>X</math> axis in 4 different places. The function has 4 roots: <math>(-2,0), (1,0), (5,0), (10,0).</math> This function contains one local minimum, one local maximum and one absolute minimum. There is no absolute maximum. Because the function contains one absolute minimum: * If abs(<math>x</math>) is very large, <math>f(x)</math> is always positive. * If absolute minimum is above <math>X</math> axis, curve does not cross <math>X</math> axis and function contains only complex roots. * There is always at least one point where the curve is parallel to <math>X</math> axis. The curve is never parallel to the <math>Y</math> axis. For any real value of <math>x</math> there is always a real value of <math>y.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0312quartic02.png|thumb|400px|'''When coefficient d is missing, there is a stationary point at x = 0.''']] If coefficient <math>d</math> is missing, the quartic function becomes <math>y = ax^4 + bx^3 + cx^2 + e,</math> and <math>y' = 4ax^3 + 3bx^2 + 2cx = x(4ax^2 + 3bx + 2c).</math> For a stationary point <math>y' = x(4ax^2 + 3bx + 2c) = 0.</math> When coefficient <math>d</math> is missing, there is always a stationary point at <math>x = 0.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} [[File:0319quartic02.png|thumb|400px|'''Graph of quartic function that is quadratic in <math>x^2</math>.'''</br>Because coefficient <math>d</math> is missing, there is a stationary point where <math>x = 0.</math>]] If coefficients <math>b, d</math> are missing, the quartic function becomes a quadratic in <math>x^2.</math> The curve (red line) in diagram has equation: <math>y = f(x) = \frac{x^4 - 13x^2 + 36}{5}</math> The quartic equation may be solved as: <math>X^2 - 13X + 36 = 0</math> where <math>X = x^2</math> or <math>x = \sqrt{X}.</math> <math>X = 4</math> or <math>X = 9.</math> <math>x = \pm 2</math> or <math>x = \pm 3.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0312quartic03.png|thumb|400px| ]] The quartic function may be expressed as <math>x = ay^4 + by^3 + cy^2 + dy + e.</math> Unless otherwise noted, references to "quartic function" on this page refer to function of form <math>y = ax^4 + bx^3 + cx^2 + dx + e.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0318quartic01.png|thumb|400px|'''Graph of quartic function with coefficient a negative.'''</br>There is no absolute minimum.]] Coefficient <math>a</math> may be negative as shown in diagram. As <code>abs</code><math>(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^4.</math> When <code>abs</code><math>(x)</math> is very large, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "quartic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0502quartic01.png|thumb|400px|'''Graph of quartic function with coefficient b missing.''' </br> Sum of roots is <math>0.</math> </br> <math>Y</math>axis compressed for clarity. ]] When sum of roots is <math>0,</math> coefficient <math>b = 0.</math> In the diagram, roots of <math>f(x)</math> are <math>-5, -4, 2, 7.</math> Sum of roots <math>= 0.</math> Therefore coefficient <math>b = 0.</math> {{RoundBoxBottom}} ==Function as product of linear function and cubic== {{RoundBoxTop|theme=3}} [[File:0313quartic01.png|thumb|400px|'''Graphs of quartic function and associated cubic function.''']] When <math>p</math> is a root of the function, the function may be expressed as: <math>(x - p)( Ax^3 + Bx^2 + Cx + D )</math> where <math>A = a;\ B = Ap + b;\ C = Bp + c;\ D = Cp + d.</math> When one real root <math>p</math> is known, the other three roots may be calculated as roots of the cubic function <math>Ax^3 + Bx^2 + Cx + D.</math> In the diagram the quartic function has equation: <math>y = \frac{ x^4 - 23x^3 + 163x^2 - 393x + 252 }{ 48 }.</math> It is known that <math>3</math> is a root of this function. The associated cubic has equation: <math>y = \frac { x^3 - 20x^2 + 103x - 84} { 48 }</math> The 2 curves coincide at points <math>(1, 0),\ (7, 0),\ (12, 0),</math> the three points that are roots of both functions. <math></math> {{RoundBoxBottom}} ==Function defined by 5 points== {{RoundBoxTop|theme=2}} [[File:0313quartic02.png|thumb|400px|'''Figure 1. Quartic function defined by 5 points''' </br>Any 5 points on the curve may be used to define the function.]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 5 unique points on the curve. For example, let us choose the five points: <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code points = (-5,0), (-2,0), (1,0), (3,-6), (6,2) L11 = [] for point in points : x,y = point L11 += [[x*x*x*x, x*x*x, x*x, x, 1, -y]] print (L11) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ 625.0, -125.0, 25.0, -5.0, 1.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 0.0], # [ 1.0, 1.0, 1.0, 1.0, 1.0, 0.0], # matrix supplied to function solveMbyN() below. [ 81.0, 27.0, 9.0, 3.0, 1.0, 6.0], # 5 rows by 6 columns. [1296.0, 216.0, 36.0, 6.0, 1.0, -2.0]] # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L11) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.02651515151515152, 0.004545454545454542, -0.847727272727273, -0.728787878787879, 1.5454545454545459) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by the 5 points <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> is <math>y = \frac{0.875 x^4 + 0.15 x^3 - 27.975 x^2 - 24.05 x + 51}{33}.</math> {{RoundBoxBottom}} ==Function defined by 3 points and 2 slopes== {{RoundBoxTop|theme=2}} [[File:0314quartic03.png|thumb|400px|'''Figure 1. Quartic function defined by 3 points and 2 slopes.''' </br>Any 3 points on the curve and the slopes at any 2 of these points may be used to define the function.</br></br>Slope at <math>(-2, -2)\ =</math> slope at <math>(6, -4)\ =\ 0.</math>]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 3 unique points on the curve and the slopes at any 2 of these points. For example, let us choose the three points: <math>(-2, -2), (6, -4), (4, 1)</math> At point <math>(-2, -2)</math> slope is <math>0.</math> At point <math>(6, -4)</math> slope is <math>0.</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> Rearrange the standard cubic function of slope to prepare for the calculation of <math>a,b,c,d,e:</math> <math> 4 x^3 a + 3 x^2 b + 2 x c + 1 d + 0 e - s = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code def makeEntry(input) : x,y,s = ( tuple(input) + (None,) )[:3] L1 = [] if s != None : L2 = [ float(v) for v in [4*x*x*x, 3*x*x, 2*x, 1, 0, -s] ] L1 += [ L2 ] L2 = [ float(v) for v in [x*x*x*x, x*x*x, x*x, x, 1, -y]] L1 += [ L2 ] return L1 t1 = ( (-2,-2, 0), # point (-2, -2) with slope 0. (6,-4, 0), # point (6, -4) with slope 0. (4,1), # point (4, 1) ) L1 = [] for v in t1 : L1 += makeEntry ( v ) print (L1) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ -32.0, 12.0, -4.0, 1.0, 0.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 2.0], # [ 864.0, 108.0, 12.0, 1.0, 0.0, 0.0], # matrix supplied to function solveMbyN() below. [1296.0, 216.0, 36.0, 6.0, 1.0, 4.0], # 5 rows by 6 columns. [ 256.0, 64.0, 16.0, 4.0, 1.0, -1.0]]. # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L1) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.03255208333333339, -0.2526041666666665, -0.3072916666666667, 2.84375, 2.375) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by three points and two slopes is: <math>y = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> ===Associated cubic functions=== {{RoundBoxTop|theme=3}} ====When p == -2==== {{RoundBoxTop|theme=1}} [[File:0314_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = -2.''' </br>In this case roots of associated cubic include x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == -2,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 15.25 x^2 + 15.75 x + 105 } {48}.</math> Three blue vertical lines show 3 values of <math>x</math> where <math>g(x) = 0</math> and <math>f(x) = f(-2)</math> In this case roots of <math>g(x)</math> include <math>x = p.</math> {{RoundBoxBottom}} ====When p == 5==== {{RoundBoxTop|theme=1}} [[File:0314_2curves02.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 5.''' </br>In this case the one root of associated cubic excludes x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 5,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 4.3125 x^2 - 36.3125 x - 45.0625 } {48}.</math> Two blue vertical lines show 2 values of <math>x</math> where <math>f(x) = f(5)</math> In this case the one root of <math>g(x)</math> excludes <math>x = p.</math> {{RoundBoxBottom}} ====When p == 6==== {{RoundBoxTop|theme=1}} [[File:0315_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 6.''' </br>In this case the one root of associated cubic includes x = p.]] Quartic function is: <math>y = f(x) </math><math>= \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 6,</math> associated cubic function is: <math>y = g(x) </math><math>= \frac{ 1.5625 x^3 - 2.75 x^2 - 31.25 x - 51 } {48}.</math> One blue vertical line shows 1 value of <math>x</math> where <math>f(x) = f(6)</math> In this case the one root of <math>g(x)</math> includes <math>x = p.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} ==Examples== {{RoundBoxTop|theme=2}} ===Quartic with 2 stationary points=== {{RoundBoxTop|theme=1}} [[File:0315_2statPoints02.png|thumb|400px|'''Graph of quartic function with 2 stationary points.''']] In the diagram the red line represents quartic function <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 2 roots of <math>g(x),\ -1</math> and <math>\frac{-1}{4}</math> show that <math>f(x)</math> has stationary points at <math>x = -1</math> and <math>x = -0.25.</math> {{RoundBoxBottom}} ===Quartic with 1 stationary point=== {{RoundBoxTop|theme=1}} [[File:0315_1statPoint.png|thumb|400px|'''Graph of quartic function with 1 stationary point.''']] In the diagram the red line represents quartic function <math>y = f(x) = \frac{x^4 + 3x^3 + 3x^2 + 3x}{2}</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 1 root of <math>g(x),\ -1.607</math> (approx.), shows that <math>f(x)</math> has 1 stationary point where <math>g(x) = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =First and second derivatives= {{RoundBoxTop|theme=1}} ==Points of inflection== {{RoundBoxTop|theme=2}} [[File:0317_3curves01.png|thumb|400px|'''Graphs of quartic function and first two derivatives.'''</br>Dotted portion of black line shows where f(x) is always concave down.</br>Dotted portion of red line shows where g(x) is decreasing.</br>Dotted portion of blue line shows where h(x) is negative.</br>When h(x) is negative, f(x) is concave down.</br>When h(x) is positive, f(x) is concave up.]] In the diagram the black line has equation: <math>y = f(x) = \frac{x^4 - 16x^3 + 42x^2 + 12x + 4}{144}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{x^3 - 12x^2 + 21x + 3}{36}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{x^2 - 8x + 7}{12}.</math> When <math>x < x_1:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. When <math>x == x_1:</math> * <math>y'</math> is at a local maximum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between up and down. When <math>x_1 < x < x_2:</math> * <math>y'</math> is decreasing. * <math>y''</math> is negative. * <math>f(x)</math> is always concave down. When <math>x == x_2:</math> * <math>y'</math> is at a local minimum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between down and up. When <math>x_2 < x:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. {{RoundBoxTop|theme=1}} The roots of <math>h(x): x_1 = 1, x_2 = 7.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At point <math>p_1</math> concavity of <math>f(x)</math> changes from up to down. At point <math>p_2</math> concavity of <math>f(x)</math> changes from down to up. The points <math>p_1, p_2</math> (the <math>X</math> coordinates of which are roots of <math>h(x)</math>) are the '''points of inflection''' of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Maxima and minima== {{RoundBoxTop|theme=2}} [[File:0317_3curves03.png|thumb|400px|'''Graphs of quartic function and first two derivatives showing maximum and minima.'''</br> Point <math>p_{2}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{2}</math> is concave down. Point <math>p_{2}</math> is local maximum. </br> Point <math>p_{1}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{1}</math> is concave up. Point <math>p_{1}</math> is local minimum. </br> Similarly, point <math>p_{3}</math> is local minimum. ]] In the diagram the black line has equation: <math>y = f(x) = \frac{1.5625x^4 - 12.125x^3 - 14.75x^2 + 136.5x + 114}{48}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{6.25x^3 - 36.375x^2 - 29.5x + 136.5}{48}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{18.75x^2 - 72.75x - 29.5}{48}.</math> Roots of <math>g(x):\ x_1 = -2;\ x_2 = 1.82;\ x_3 = 6.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> At <math>x_1\ h(x_1)</math> is positive. Point <math>p_1</math> is a stationary point and <math>f(x)</math> at <math>p_1</math> is concave up. Point <math>p_1</math> is a local minimum. Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At <math>x_2\ h(x_2)</math> is negative. Point <math>p_2</math> is a stationary point and <math>f(x)</math> at <math>p_2</math> is concave down. Point <math>p_2</math> is a local maximum. Let point <math>p_3</math> on <math>f(x)</math> have coordinates <math>(x_3, f(x_3)).</math> At <math>x_3\ h(x_3)</math> is positive. Point <math>p_3</math> is a stationary point and <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is a local minimum. {{RoundBoxBottom}} ==Quartic with 2 stationary points== {{RoundBoxTop|theme=3}} [[File:0318_3curves01.png|thumb|400px|'''Graph of quartic function with 2 stationary points and first 2 derivatives.''' </br> Black line: <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> </br> Red line: <math>y' = g(x) = 4(4x^3 + 9x^2 + 6x + 1).</math> </br> <math>y'' = h(x) = 4(12x^2 + 18x + 6)</math> </br> Blue line: <math>y'' = 4(2x^2 + 3x + 1) = \frac{h(x)}{6}.</math> </br> Dotted portion of black line shows where <math>f(x)</math> is concave down. ]] In the diagram, point <math>p_1</math> on <math>f(x)</math> has coordinates <math>(x_1, f(x_1)).</math> Similarly, points <math>p_2, p_3</math> have coordinates <math>(x_2, f(x_2)),\ (x_3, f(x_3)).</math> <math>y'</math> has roots: <math>x_1 = -1;\ x_3 = -0.25.</math> Points <math>p_1, p_3</math> are stationary points. <math>y''</math> has roots: <math>x_1 = -1;\ x_2 = -0.5.</math> Points <math>p_1, p_2</math> are points of inflection. At point <math>p_3\ y''</math> is positive. <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is local minimum. {{RoundBoxTop|theme=2}} Summary: * Point <math>p_1</math> is both stationary point and point of inflection. * Point <math>p_2</math> is point of inflection. * Point <math>p_3</math> is both stationary point and local minimum. {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =The simplest quartic function= {{RoundBoxTop|theme=2}} [[File:0320quartic01.png|thumb|400px|'''Graph of simplest quartic function.''' </br> Point <math>p_1\ (-1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_2\ (1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_0\ (0, -1.1^4)</math> is <math>Y</math> intercept of <math>f(x).</math> ]] The simplest quartic function has coefficients <math>b = c = d = 0.</math> Red line in diagram has equation: <math>y = f(x) = x^4 - 1.1^4</math> First derivative (not shown) of <math>f(x):\ y' = g(x) = 4x^3.</math> When <math>x == 0,\ g(x) = 0.</math> There is a stationary point on <math>f(x)</math> when <math>x == 0,</math> point <math>p_0.</math> Second derivative (not shown) of <math>f(x):\ y'' = h(x) = 12x^2.</math> When <math>x == 0,\ h(x) = 0.</math> There is a point of inflection on <math>f(x)</math> when <math>x == 0.</math> For every non-zero value of <math>x,\ h(x)</math> is positive. To left and right of point <math>p_0,\ f(x)</math> is always concave up. Point <math>p_0</math> is both local minimum and absolute minimum. * Point <math>p_0</math> is stationary point and point of inflection and absolute minimum. Curve <math>f(x)</math> is useful for finding the fourth root of a real number. Solve: <math>x = N^{\frac{1}{4}}.</math> <math>x^4 = N.</math> <math>x^4 - N = 0.</math> This is equivalent to finding a root of function <math>y = j(x) = x^4 - N.</math> If you use Newton's method to find a root of <math>j(x),</math> this would be more efficient than solving <math>x = \sqrt{\sqrt{N}}.</math> {{RoundBoxBottom}} =Roots of equal absolute value= {{RoundBoxTop|theme=1}} The standard quartic function: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> For <math>x</math> in <math>(1)</math> substitute <math>(p+q).</math> Call this <math>(2).</math> For <math>x</math> in <math>(1)</math> substitute <math>(p-q).</math> Call this <math>(3).</math> Combine <math>(2)</math> and <math>(3)</math> to eliminate <math>q</math> and produce an equation in <math>p:</math> <math>(- 64aaa)pppppp+</math> <math>(- 96aab)ppppp+</math> <math>(- 32aac - 48abb)pppp+</math> <math>(- 32abc - 8bbb)ppp+</math> <math>(+ 16aae - 4abd - 4acc - 8bbc)pp+</math> <math>(+ 8abe - 2bbd - 2bcc)p+</math> <math>(+ add + bbe - bcd)\ =\ 0\ \dots\ (4).</math> We are interested in coefficient <math>0</math> of <math>(4):\ c_0 = add + bbe - bcd.</math> If <math>c_0 == 0,\ p=0</math> is a solution and function <math>(1)</math> has 2 roots of form <math>0 \pm q</math> where <math>q = \sqrt{\frac{-d}{b}}.</math> An example: {{RoundBoxTop|theme=3}} [[File:0320quartic02.png|thumb|400px|'''Graph of quartic function with 2 roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 12x^3 + 31x^2 + 48x -140}{45}.</math> <math>a,b,c,d,e = 1,-12,31,48,-140</math> <math>c_0 = add + bbe - bcd </math><math>= 1(48)(48) + (-12)(-12)(-140) - (-12)(31)(48) = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-48}{-12}} = \sqrt{4} = \pm 2.</math> The 2 roots of equal absolute value are: <math>2, -2.</math> {{RoundBoxBottom}} The method works with complex roots of equal absolute value: {{RoundBoxTop|theme=2}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with 2 complex roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}.</math> <math>a,b,c,d,e = 1,-3,-1,-27,-90</math> <math>c_0 = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-(-27)}{-3}} = \sqrt{-9} = \pm 3i.</math> The 2 roots of equal absolute value are: <math>3i, -3i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Equal roots= {{RoundBoxTop|theme=2}} Equal roots occur when the function and the slope of the function both equal zero. <math>ax^4 + bx^3 + cx^2 + dx + e = 0\ \dots\ (1)</math> <math>4ax^3 + 3bx^2 + 2cx + d = 0\ \dots\ (2)</math> Begin the process of reducing <math>(1),\ (2)</math> to linear functions. Combine <math>(1),\ (2)</math> to produce 2 cubic functions: <math>Fx^3 + Gx^2 + Hx + J\ \dots\ (1a)</math> where: <math>F = ad;\ G = bd - 4ae;\ H = cd - 3be;\ J = dd - 2ce.</math> <math>fx^3 + gx^2 + hx + j\ \dots\ (2a)</math> where: <math>f = 4a;\ g = 3b;\ h = 2c;\ j = d.</math> Combine <math>(1a),\ (2a)</math> to produce 2 quadratic functions: <math>Kx^2 + Lx + M\ \dots\ (1b)</math> where: <math>K = Gf -Fg;\ L = Hf -Fh;\ M = Jf - Fj.</math> <math>kx^2 + lx + m\ \dots\ (2b)</math> where: <math>k = Fj - Jf;\ l = Gj - Jg;\ m = Hj - Jh.</math> Combine <math>(1b),\ (2b)</math> to produce 2 linear functions: <math>Rx + S\ \dots\ (1c)</math> where: <math>R = Lk - Kl;\ S = Mk - Km.</math> <math>rx + s\ \dots\ (2c)</math> where: <math>r = Km-Mk;\ s = Lm - Ml.</math> From <math>(1c):\ x_1 = \frac{-S}{R}</math> From <math>(2c):\ x_2 = \frac{-s}{r}</math> If <math>x_1 == x_2:</math> <math>\frac{-S}{R} = \frac{-s}{r}</math> <math>Rs = rS</math> <math>Rs - Sr = 0.</math> The value <math>Rs - Sr</math> is in fact: <syntaxhighlight> + 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee - 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd + 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee + 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde + 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee - 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde - 3aabbccdddddd </syntaxhighlight> which, by removing values <math>aa, d</math><math>d</math> (common to all values), may be reduced to: <syntaxhighlight> status = ( + 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee - 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd + 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee + 384abccddde - 54abcddddd + 128acccccee - 80accccdde + 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee - 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde - 3bbccdddd ) </syntaxhighlight> If <math>status == 0,</math> there are at least 2 equal roots which may be calculated as shown below. {{RoundBoxTop|theme=2}} If coefficient <math>d</math> is non-zero, it is not necessary to calculate <math>status.</math> If coefficient <math>d == 0,</math> verify that <math>status = 0</math> before proceeding. {{RoundBoxBottom}} ==No equal roots== {{RoundBoxTop|theme=1}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with no equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}</math> <math>a,b,c,d,e = 1, -3, -1, -27, -90</math> <math>R, S = 15269148, -35977608</math> <math>x_1 = \frac{-S}{R} = \frac{35977608}{15269148} = 2.3562289133617\dots</math> <math>r, s = 35977608, -60634332</math> <math>x_2 = \frac{-s}{r} = \frac{60634332}{35977608} = 1.685335278543253\dots</math> <math>x_1 != x_2.</math> There are no equal roots. {{RoundBoxBottom}} ==Exactly 2 equal roots== {{RoundBoxTop|theme=1}} [[File:0329quartic04.png|thumb|400px|'''Graph of quartic function with exactly 2 equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 + 6x^3 - 48x^2 - 182x + 735}{100}</math> <math>a,b,c,d,e = 1, 6, -48, -182, 735</math> <math>R, S = -1027353600, -7191475200</math> <math>x_1 = \frac{-S}{R} = \frac{7191475200}{-1027353600} = -7</math> <math>r, s = 7191475200, 50340326400</math> <math>x_2 = \frac{-s}{r} = \frac{-50340326400}{7191475200} = -7</math> <math>x_1 = x_2 = -7.</math> There are 2 equal roots at <math>x = -7.</math> {{RoundBoxTop|theme=2}} The following 3 graphs show the steps that lead to calculation of equal roots at point <math>(-7, 0).</math> <gallery> File:0914quartic00.png|<small>Figure 1. graph of <math>f(x)</math> and 2 associated cubic functions.</small> File:0914quartic01.png|<small>Figure 2. graph of <math>f(x)</math> and 2 associated quadratic functions.</small> File:0914quartic02.png|<small>Figure 3. graph of <math>f(x)</math> and 2 associated linear functions.</small> </gallery> In all graphs, all curves have a common root at point <math>(-7, 0).</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=4}} See [[Quartic_function#Function_as_product_of_linear_function_and_cubic | "Function as product of linear function and cubic"]] above. To calculate all roots: <syntaxhighlight lang=python> # python code. a,b,c,d,e = 1,6,-48,-182,735 # The associated cubic: p = -7 A = a B = A*p + b C = B*p + c D = C*P + d # The associated quadratic: a1 = A b1 = a1*p + B c1 = b1*p + C a1,b1,c1 </syntaxhighlight> <syntaxhighlight> (1, -8, 15) </syntaxhighlight> Roots of quadratic function <math>g(x) = x^2 - 8x + 15</math> are <math>3, 5.</math> All roots of <math>f(x)</math> are <math>-7, -7, 3, 5.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Exactly 3 equal roots== {{RoundBoxTop|theme=1}} [[File:0320_2curves01.png|thumb|400px|'''Graph of quartic function with exactly 3 equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 2x^3 - 36x^2 + 162x - 189}{100}</math> <math>a,b,c,d,e = 1, -2, -36, 162, -189</math> <math>R, S = 0, 0\ \dots\dots\ (1c)</math> <math>r, s = 0, 0\ \dots\dots\ (2c)</math> In this case the calculation of <math>x_1, x_2</math> is not appropriate because there are more than 2 equal roots. Try equations <math>(1b), (2b).</math> Both of these are equivalent to: <math>y = g(x) = x^2 - 6x + 9,</math> blue line in diagram. Discriminant of <math>g(x) = (-6)^2 - 4(1)(9) = 0.\ g(x)</math> has two equal roots at <math>x = \frac{-(-6)}{2(1)} = 3.</math> Therefore <math>f(x)</math> has 3 equal roots at <math>x = 3.</math> {{RoundBoxBottom}} ==Four equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves01.png|thumb|400px|'''Graph of quartic function with 4 equal roots and corresponding cubic.''' </br> <math>g(x) = \frac{f'(x)}{4}.</math> ]] Red line in diagram is of function: <math>y = f(x) = x^4 - 20x^3 + 150x^2 - 500x + 625.</math> <math>a,b,c,d,e = 1, -20, 150, -500, 625</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> <math>K, L, M = 0, 0, 0</math> <math>k, l, m = 0, 0, 0</math> In this case <math>(1b), (2b), (1c), (2c)</math> are all null. This is the only case in which <math>(1b), (2b)</math> are null. <math>(1a), (2a)</math> are both equivalent to: <math>y = g(x) = x^3 - 15x^2 + 75x - 125,</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 4 equal roots at <math>x = 5.</math> {{RoundBoxBottom}} ==Two pairs of equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 6x^3 - 11x^2 + 60x + 100}{20}.</math> <math>a,b,c,d,e = 1, -6, -11, 60, 100</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> In this case <math>(1c), (2c)</math> are both null. <math>(1b), (2b)</math> are both equivalent to: <math>y = g(x) = \frac{x^2 - 3x - 10}{20},</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = -2</math> and one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 2 equal roots at <math>x = -2</math> and 2 equal roots at <math>x = 5.</math> {{RoundBoxTop|theme=2}} [[File:0408_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal, complex roots and corresponding quadratic.''' </br> <math>g(x) = (x-(3-2i))(x-(3+2i)) = x^2 - 6x + 13.</math> </br> <math>f(x) = (g(x))^2.</math> ]] This method is valid for complex roots. For example: <math>y = f(x) = x^4 - 12x^3 + 62x^2 - 156x + 169.</math> <math>a,b,c,d,e = 1,-12,62,-156,169.</math> In this case <math>(1c),\ (2c)</math> are both null. <math>(1b),\ (2b)</math> are both equivalent to: <math>y = g(x) = x^2 - 6x + 13,</math> blue line in diagram. Roots of <math>g(x)</math> are <math>3 \pm 2i.</math> <math>f(x)</math> has 2 roots equal to <math>3 + 2i</math> and 2 roots equal to <math>3 - 2i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Summmary== {{RoundBoxTop|theme=1}} {| class="wikitable" |- ! || No equal roots !! 2 equal roots !! 3 equal roots !! 4 equal roots !! 2 pairs of equal roots |- | Cubic: 1(a), 2(a) | different | different | different | same | different |- | Quadratic: 1(b), 2(b) | different | different | same, 1root | null | same, 2roots |- | Linear: 1(c), 2(c) | different | same | null | null | null |} {{RoundBoxBottom}} ==Caution== {{RoundBoxTop|theme=4}} [[File:0915quartic02.png|thumb|400px| Calculation of false equal roots.</br> In this example, method calculates 2 legitimate equal roots at <math>(4, 0)</math> and 2 false equal roots at <math>(7.543296089385474,0).</math> <math></math> ]] Black line in diagram has equation: <math>y = f(x) = 0.012684240362811794x^4</math> <math> -\ 0.19522392290249435x^3 </math> <math> +\ 0.7654478458049887x^2 + 0x - 3. </math> <math>f(x)</math> is a quartic function with exactly 2 equal roots and coefficient <math>d</math> missing. Calculation of equal roots of <math>f(x)</math> gives linear functions <math>(1c), (2c)</math> null and quadratic functions <math>(1b), (2b)</math> with equal roots of <math> (4,0), (7.543296089385474,0). </math> Usually, this indicates that <math>f(x)</math> should have 2 equal roots at <math>(4,0)</math> and 2 equal roots at <math>(7.543296089385474,0).</math> It is obvious that <math>7.543296089385474</math> is not a root of <math>f(x).</math> When <math>x = 7.543296089385474,</math> slope of derivative <math>g(x) = 0.</math> Value of <math>f(x)\ != 0.</math> This example indicates that it would be wise to verify that calculated equal roots are in fact valid roots of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Depressed quartic= {{RoundBoxTop|theme=1}} A depressed quartic is any quartic function with any one or more of coefficients <math>b,c,d</math> missing. Within this section a depressed quartic has coefficient <math>b</math> missing. To produce the depressed quartic: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> <math>y = \frac{(4^4 a^3)(ax^4 + bx^3 + cx^2 + dx + e)}{4^4 a^3}\ \dots\ (2)</math> Let <math>x = \frac{-b + t}{4a}.</math> Substitute in <math>(2),</math> expand and simplify: <math>y = \frac{t^4 + At^2 + Bt + C}{4^4a^3}\ \dots\ (3)</math> where: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> When equated to <math>0,\ (3)</math> becomes the depressed equation: <math>t^4 + At^2 + Bt + C = 0\ \dots\ (4).</math> Be prepared for the possibility that any 1 or more of <math>A, B, C</math> may be zero. ==Coefficient B missing== {{RoundBoxTop|theme=2}} [[File:0322_3curves01.png|thumb|400px|'''Graph of quartic function that resembles a quadratic.''' ]] If coefficient <math>B == 0,\ (4)</math> becomes a quadratic in <math>t^2:</math> <math>t^4 + At^2 + C = 0.</math> <math>(1)</math> has the appearance of a quadratic. The black line: <math>y = f(x) = \frac{x^4 - 4x^3 + 9x^2 - 10x + 5}{10}</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>= 8( 8(-10) - 4(-4)(9) + -64 )</math> <math>= 8(-80 + 144 - 64)</math> <math>= 8(0) = 0.</math> The red line: <math>y' = g(x) = \frac{4x^3 - 12x^2 + 18x - 10}{10}</math> <math>y' = g(x) = 0</math> where <math>x = 1.</math> The grey line: <math>y'' = h(x) = \frac{12x^2 - 24x + 18}{10}</math> * Absolute minima of <math>f(x)</math> and of <math>h(x)</math> and point of inflection of <math>g(x)</math> occur where <math>x = \frac{-b}{4} = 1.</math> * <math>y''</math> is always positive. <math>f(x)</math> is always concave up. * <math>f(1 + p) = f(1-p).</math> {{RoundBoxTop|theme=3}} If <math>(1)</math> contains 2 pairs of equal roots, coefficient <math>B = 0.</math> The converse is not necessarily true. If <math>(1)</math> contains 4 equal roots, coefficients <math>A = B = C = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Coefficient C missing== {{RoundBoxTop|theme=2}} [[File:0501quartic01.png|thumb|400px|'''Graph of quartic function with coefficient C of depressed function missing.''' </br> <math>Y</math> axis compressed for clarity. ]] If coefficient <math>C == 0,\ (4)</math> becomes: <math>t^4 + At^2 + Bt = t(t^3 + At + B) = 0</math> in which case <math>t = 0</math> is a solution and <math>x = \frac{-b}{4a}</math> is a root of <math>(1).</math> Curve (red line) in example has equation: <math>y = f(x) = 8x^4 + 16x^3 + 24x^2 + 89x + 40.</math> Coefficients of depressed function are: <syntaxhighlight lang=python> # python code a,b,c,d,e = 8, 16, 24, 89, 40 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b A,B,C </syntaxhighlight> <syntaxhighlight> (1536, 299008, 0) </syntaxhighlight> Coefficient <math>C</math> of depressed function is missing. <math>t = 0</math> is a solution. Using <math>x = \frac{-b + t}{4a},</math> one root of <math>f(x) = \frac{-16 + 0}{4(8)} = -0.5.</math> {{RoundBoxBottom}} ==Resolvent cubic== {{RoundBoxTop|theme=1}} This section introduces a special cubic function called "resolvent" because it helps to resolve a requirement, the calculation of the roots of the quartic. The depressed quartic: <math>t^4 + At^2 + Bt + C\ \dots\ (1)</math> For <math>t</math> substitute <math>(u+v)\ \dots\ (2)</math> For <math>t</math> substitute <math>(u-v)\ \dots\ (3)</math> <math>(2)+(3):\ 2Auu + 2Avv + 2Bu + 2C + 2uuuu + 12uuvv + 2vvvv\ \dots\ (4)</math> Simplify <math>(4):\ Auu + AV + Bu + C + uuuu + 6uuV + VV\ \dots\ (4a)</math> <math>(2)-(3):\ 4Auv + 2Bv + 8uuuv + 8uvvv\ \dots\ (5)</math> Simplify <math>(5):\ 2Au + B + 4uuu + 4uV\ \dots\ (5a)</math> From <math>(5a):\ 4uV = -(2Au + B + 4uuu)\ \dots\ (5b)</math> <math>(4a)*4u4u:\ 4u4uAuu + A4u(4uV) + 4u4uBu + 4u4uC + 4u4uuuuu + 6uu4u(4uV) + (4uV)(4uV)\ \dots\ (6)</math> In <math>(6)</math> replace <math>4uV</math> with <math>(-(2Au + B + 4uuu)),</math> expand, simplify, gather like terms and result is: <math>Pu^6 + Qu^4 + Ru^2 + S</math> or {{RoundBoxTop|theme=2}} <math>PU^3 + QU^2 + RU + S\ \dots\ (7)</math> where: <math>U = u^2</math> <math>P = 64</math> <math>Q = 32A</math> <math>R = 4A^2 - 16C</math> <math>S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} From <math>(5b):\ V = v^2 = \frac{-(2Au + B + 4uuu)}{4u} = -(\frac{A}{2} + U) - \frac{B}{4u}\ \dots\ (8)</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Some simple changes reduce the number of calculations and also the sizes of coefficients <math>P, Q, R, S.</math> <math>A2 = 8ac - 3b^2</math> where <math>A2 = \frac{A}{2}</math> <math>B4 = 16a^2 d - 8abc + 2b^3</math> where <math>B4 = \frac{B}{4}</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> Then: <math>P = 64</math> <math>Q = 32(A2)(2) = 64A2</math> <math>R = 4(A2)(2)(A2)(2) - 16C = 16A2^2 - 16C</math> <math>S = -(B4)(4)(B4)(4) = -16 B4^2</math> Divide all 4 coefficients by <math>16:</math> <math>P = 4</math> <math>Q = 4A2</math> <math>R = A2^2 - C</math> <math>S = -B4^2</math> <math>V = v^2 = -(A2 + U) - \frac{B4}{u}.</math> A close examination of coefficients <math>R, S</math> shows that both coefficients are always exactly divisible by <math>4.</math> Therefore, all four coefficients may be defined as follows: <math>P = 1</math> <math>Q = A2</math> <math>R = \frac{A2^2 - C}{4}</math> <math>S = \frac{-B4^2}{4}</math> Actual calculations of values <math>R, S</math> are: <syntaxhighlight lang=python> # python code. R = ( (16)*(a)*(a)*(b)*(d) + (16)*(a)*(a)*(c)*(c) + (3)*(b)*(b)*(b)*(b) + (-16)*(a)*(b)*(b)*(c) + (-64)*(a)*(a)*(a)*(e) ) S = ( (64)*(a)*(a)*(a)*(b)*(c)*(d) + (8)*(a)*(b)*(b)*(b)*(b)*(c) + (-16)*(a)*(a)*(b)*(b)*(b)*(d) + (-16)*(a)*(a)*(b)*(b)*(c)*(c) + (-1)*(b)*(b)*(b)*(b)*(b)*(b) + (-64)*(a)*(a)*(a)*(a)*(d)*(d) ) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =Solving quartic equation= This section presents 4 examples that show how to use the depressed quartic and the resolvent cubic to solve the quartic equation. ==Four real roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves01.png|thumb|400px|'''Graphs of quartic function with 4 real roots and associated resolvent cubic.''' </br>Resolvent cubic contains 3 real, positive roots. ]] Calculate roots of: <math>y = f(x) = x^4 - x^3 - 19x^2 - 11x + 30</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,-1,-19,-11,30 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -310 -1320 6669 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 -9920 277696 -1742400 </syntaxhighlight> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = 64x^3 - 9920x^2 + 277696x - 1742400.</math> There are 3 real, positive roots: <math>9, 25, 121.</math> Using 3 roots of <math>g(x),</math> calculate 4 roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code for U in (9, 25, 121) : print ('\nU =', U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** .5 for t in (u+v, u-v) : x = (-b+t) / (4*a) y = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x:',x, '; y:',y) </syntaxhighlight> <syntaxhighlight> U = 9 x: 5.0 ; y: 0.0 x: -3.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 U = 25 x: 5.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -3.0 ; y: 0.0 U = 121 x: 5.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: -3.0 ; y: 0.0 </syntaxhighlight> Roots of <math>f(x)</math> are: <math>5, 1, -2, -3.</math> {{RoundBoxBottom}} All 3 values of <math>U</math> produce the same results, but not in same sequence. It is not necessary to calculate all 3 roots of resolvent cubic. Any one non-zero root is sufficient to do the job. ==Two real and two complex roots== ===Example 1=== {{RoundBoxTop|theme=4}} [[File:0323_2curves02.png|thumb|400px|'''Graphs of quartic function with 2 real and 2 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 + 2x^3 + 18x^2 - 70x - 87</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,2,18,-70,-87 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> 264 -5568 -12208 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 8448 474112 -31002624 </syntaxhighlight> {{RoundBoxBottom}} Calculate one real root of cubic function: <math>y = g(x) = 64x^3 + 8448x^2 + 474112x -31002624.</math> <math>36</math> is one real root. Choose <math>U = 36.</math> Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 36 u1 = U**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) if V >= 0 : v = V**.5 else : v = 1j * (-V)**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =',x, '; sum =',sum ) </syntaxhighlight> <syntaxhighlight> x = 3.0 ; sum = 0.0 x = -1.0 ; sum = 0.0 x = (-2+5j) ; sum = 0j x = (-2-5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=4}} [[File:0321_2curves00.png|thumb|400px| Coefficient <math>d</math> of <math>g(x) = 0.</math> <math></math> ]] Calculate roots of: <math>y = f(x)</math> <math>= 3x^4 - 6x^3 - 41x^2 +44x - 189</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 3, -6, -41, 44, -189 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -2184, 0, -1229040 </syntaxhighlight> Notice that coefficient <math>B = 0.</math> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 1, -1092, 605376, 0 </syntaxhighlight> Notice that coefficient <math>S = 0.</math> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = x^3 - 1092x^2 + 605376x + 0.</math> Roots are <math>0, 546 \pm 554.3103823671355j.</math> Value <math>0</math> cannot be used because it will cause error <code>Divide by zero</code> at statement <code>V = -(A/2 + U) - B/(4*u)</code>. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 546+554.3103823671355j print ('\nU =',U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** 0.5 s1 = '\nu,v' print (s1,eval(s1)) for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> U = (546+554.3103823671355j) u,v ((25.729935131257832+10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (4.788322521876306 + 0j) ; sum = (1.9895196601282805e-13 + 0j) x = (0.5 + 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) u,v ((-25.729935131257832-10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (0.5 - 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) x = (-3.7883225218763052 + 0j) ; sum = (1.7053025658242404e-13 + 0j) </syntaxhighlight> Values of <math>x</math> are: <math>-3.7883225218763052, 4.788322521876306, 0.5 \pm 1.795283650280614j</math> <math></math><math></math> {{RoundBoxBottom}} ====Depressed quartic as quadratic==== In this example coefficient <math>B</math> of depressed quartic <math>= 0.</math> Therefore, resolvent cubic can be ignored and depressed quartic processed as quadratic in <math>T = t^2.</math> <math>t^4 - 2184t^2 + (0)t - 1229040</math> <math>T^2 - 2184T - 1229040</math> where <math>T = t^2.</math> Solutions of this quadratic are: <math>T_1, T_2 = 2648.1182474349434, -464.11824743494344</math> <syntaxhighlight lang=python> T1,T2 = 2648.1182474349434, -464.11824743494344 t1 = T1 ** 0.5; t2 = ((-T2) ** 0.5) * 1j for t in (t1,-t1,t2,-t2) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> x = 4.788322521876305 ; sum = -1.7053025658242404e-13 x = -3.788322521876305 ; sum = -1.7053025658242404e-13 x = (0.5 + 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) x = (0.5 - 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) </syntaxhighlight> or <syntaxhighlight> x = 0.5 ± 4.288322521876305, 0.5 ± 1.7952836502806138j </syntaxhighlight> With precision of 15, values of <math>x</math> are same as those shown above. When roots of quartic function are of form <code>p ± q, p ± r,</code> coefficient <math>B</math> of depressed function <math>= 0.</math> ==Four complex roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves03.png|thumb|400px|'''Graphs of quartic function with 4 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 - 20x^3 + 408x^2 + 2296x + 18020</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 4128 344064 9683200 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 64 132096 -86769664 -118380036096 </syntaxhighlight> {{RoundBoxBottom}} Calculate one root of cubic function: <math>y = g(x) = 64x^3 + 132096x^2 - 86769664x - 118380036096.</math> There are 3 real roots: <math>-2304, -784, 1024.</math> Choose <math>U = -784.</math> Negative <math>U</math> is chosen here to show that any 1 of the roots produces the correct result. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = -784 u1 = 1j * (-U)**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) v = V**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> # python expresses complex numbers with 'j'. x = (13+19j) ; sum = 0j x = (-3-5j) ; sum = 0j x = (13-19j) ; sum = 0j x = (-3+5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} =Quartic formula= The substitutions made above can be used to produce a formula for the solution of the quartic equation. See main articles [https://en.wikipedia.org/wiki/Quartic_equation#The_general_case "The general case" ] or [https://en.wikipedia.org/wiki/Quartic_function#General_formula_for_roots "General formula for roots."] {{RoundBoxTop|theme=2}} Both links above point to formula for equation <math>x^4 + ax^3 + bx^2 + cx + d = 0.</math> {{RoundBoxBottom}} Given quartic equation: <math>ax^4 + bx^3 + cx^2 + dx + e = 0,</math> calculate the 4 values of <math>x.</math> <math>x = \frac{-b + t}{4a}</math> where: {{RoundBoxTop|theme=2}} Coefficients of depressed quartic: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of resolvent cubic: <math>a_1 = P = 64</math> <math>b_1 = Q = 32A</math> <math>c_1 = R = 4A^2 - 16C</math> <math>d_1 = S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of depressed cubic: <math>A_1 = 9a_1 c_1 - 3b_1^2</math> <math>B_1 = 27a_1^2d_1 - 9a_1b_1c_1 + 2b_1^3</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of resolvent cubic: <math>C_1 = \frac{-A_1}{3} = b_1^2 - 3a_1 c_1</math> <math>\Delta = B_1^2 - 4C_1^3\ \dots\dots\ \Delta</math> may be negative. <math>\delta = \sqrt{\Delta}</math> <math>W = \frac{-B_1 + \delta}{2}</math> <math>w = \sqrt[3]{W}</math> <math>t_1 = w + \frac{C_1}{w}</math> <math>U = \frac{-b_1 + t_1}{3a_1}</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of quartic: <math>u = \sqrt{U}\ \dots\dots\ u</math> may be positive or negative. <math>V = -(\frac{A}{2} + U) - \frac{B}{4u}</math> <math>v = \sqrt{V}\ \dots\dots\ v</math> may be positive or negative. <math>t = u + v</math> {{RoundBoxBottom}} Formula above produces one value of <math>x.</math> Python code below utilizes <math>\pm \sqrt{U}</math> and <math>\pm \sqrt{V}</math> to produce 4 values of <math>t</math> and then, four values of <math>x.</math> ==An example:== {{RoundBoxTop|theme=4}} [[File:0330quartic01.png|thumb|400px|'''Graph of quartic function with 2 real roots.'''</br><math>Y</math> axis compressed for clarity.]] Calculate roots of <math>f(x) = 4x^4 + 4x^3 - 75x^2 - 776x - 1869.</math> <syntaxhighlight lang=python> # Python code. a,b,c,d,e = 4, 4, -75, -776, -1869 values_of_t = [ t # Coefficients of depressed quartic: for A in (16*a*c - 6*b*b,) for B in (64*a*a*d -32*a*b*c + 8*b*b*b,) for C in (256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b,) # Coefficients of resolvent cubic: for a1 in (64,) for b1 in (32*A,) for c1 in (4*A*A - 16*C,) for d1 in (-B*B,) for U in [ # The resolvent cubic: (-b1+t1)/(3*a1) # Coefficients of depressed resolvent cubic: for A1 in (9*a1*c1 - 3*b1*b1,) for B1 in (27*a1*a1*d1 - 9*a1*b1*c1 + 2*b1*b1*b1,) # One root of resolvent cubic: for C1 in (-A1/3,) for Δ in (B1*B1 - 4*C1*C1*C1,) for δ in (Δ**0.5,) for W in ((-B1 + δ)/2,) for w in (W**(1/3),) for t1 in (w + C1/w,) # See note below. ] # Prepare to calculate 4 values of t. for u1 in (U**.5,) for v1 in ( -(A/2 + U) ,) # Calculate 4 values of t. for u in (u1, -u1,) for V in ( v1 - B/(4*u),) for v in (V**.5,) for t in (u+v, u-v) ] print ('values_of_t =', values_of_t) </syntaxhighlight> {{RoundBoxTop|theme=4}} <syntaxhighlight> values_of_t = [116, -44, (-36+64j), (-36-64j)] </syntaxhighlight> Because <math>f(t)</math> is a depressed quartic function, sum of four <code>values_of_t</code> <math>= 116 - 44 - 36(2) = 0.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> # Python code. # Calculate 4 separate roots. values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] print ('values_of_x =', values_of_x) </syntaxhighlight> <syntaxhighlight lang=python> values_of_x = [7, -3, (-2.5 + 4j), (-2.5 - 4j)] </syntaxhighlight> In python the imaginary part of a complex number is shown with <math>j</math> instead of <math>i.</math> If <math>A == B == C == 0,\ f(x)</math> contains 4 equal roots and <math>x = \frac{-b}{4a}.</math> If <math>f(x)</math> contains 3 or more equal roots, statement <code>for t1 in (w + C1/w,)</code> fails with divisor <math>w = 0.</math> Before using this formula, check for equal roots as in [[Quartic_function#Exactly_3_equal_roots | "Exactly 3 equal roots"]] above. {{RoundBoxTop|theme=1}} Values displayed above have been edited slightly. Actual calculated values were: <syntaxhighlight> values_of_x = [7.000000000000001, -3.0000000000000044, (-2.499999999999998+4.000000000000001j), (-2.4999999999999987-4.000000000000001j)] </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ==In practice== {{RoundBoxTop|theme=5}} The following Python code implements the quartic formula. However, under statement <code>if B4 == 0 :</code> there is code that processes the depressed quartic as a quadratic in <math>T = t^2.</math> This ensures that execution of formula does not fail with error <code>Divide by zero</code> at statement <code>for V in ( v1 - B4/u,).</code> {{RoundBoxTop|theme=4}} <syntaxhighlight lang=python> # python code import cmath cxSqrt = cmath.sqrt # Square root of complex number. def rootsOfQuartic (abcde) : ''' x1,x2,x3,x4 = rootsOfQuartic ((a,b,c,d,e)) Each member of input must be int or float or Decimal object. Int or Decimal object in input is quietly converted to float. Output may be None. ''' def formatResults (x1x2x3x4) : ''' This function improves appearance of results. (8 + 0j) becomes 8.0 ''' values_of_x = list (x1x2x3x4) for p in (0,1,2,3) : v = values_of_x[p] if isinstance (v, complex) and (v.imag == 0) : values_of_x[p] = v.real return values_of_x status = 0 try : a,b,c,d,e = [ float(v) for v in abcde ] except : status = 1 if status : print ('rootsOfQuartic () 1: Error creating coefficients a,b,c,d,e.') return None if a == 0 : print ('rootsOfQuartic () 2: Coefficient a must be non-zero.') return None # Coefficients of depressed quartic, modified. # A = 16*a*c - 6*b*b A2 = 8*a*c - 3*b*b # B = 64*a*a*d - 32*a*b*c + 8*b*b*b B4 = 16*a*a*d - 8*a*b*c + 2*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b if B4 == 0 : # B = 0. # Result returned from this section is type tuple, indicating that coefficient B4 = 0. if A2==C==0 : # 4 equal roots. root = -b/(4*a) return tuple(formatResults((root,root,root,root))) # t**4 + At**2 + (0)t + C # Depressed quartic is quadratic in T: # T**2 + AT + C where T = t**2 # T**2 + 2(A2)T + C where A = 2(A2) # -2(A2) +/- (4(A2)(A2) - 4C)**0.5 # T = -------------------------------- = -A2 +/- ((A2)(A2) - C)**0.5 # 2 disc = A2*A2 - C if disc >= 0 : root = disc ** 0.5 else : root = ((-disc) ** 0.5) * 1j T1 = -A2 - root ; T2 = -A2 + root t1 = cxSqrt(T1) ; t2 = cxSqrt(T2) values_of_t = (t1,-t1,t2,-t2) values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] return tuple(formatResults(values_of_x)) # B4 is non-zero. Therefore, all of (S, U, u) are non-zero. P,Q,R,S = 1, A2, (A2*A2 - C)/4, -B4*B4/4 # str1 = 'P,Q,R,S' ; print (str1, eval(str1)) U = oneRootOfCubic((P,Q,R,S)) # Resolvent cubic. if U > 0 : sqrtU = U ** 0.5 elif U == 0 : # This should not happen. print ('rootsOfQuartic () 3: Internal error.') return None else : sqrtU = ((-U) ** 0.5) * 1j v1 = -(A2+U) values_of_t = [ t for u in (sqrtU, -sqrtU) for V in ( v1 - B4/u, ) for v in ( cxSqrt(V), ) for t in (u+v, u-v) ] values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] # Result returned from this section is type list, indicating that coefficient B4 != 0. return formatResults(values_of_x) </syntaxhighlight> For function <code>oneRootOfCubic()</code> see [[Cubic_function#In_practice | Cubic_function: In_practice.]] {{RoundBoxBottom}} ===Examples=== {{RoundBoxTop|theme=2}} Python function <code>equalRoots()</code> below implements <code>status</code> as presented under [https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above. <syntaxhighlight lang=python> # python code def equalRoots(abcde) : ''' This function returns True if quartic function contains at least 2 equal roots. ''' a,b,c,d,e = abcde aa = a*a ; aaa = aa*a bb = b*b ; bbb = bb*b ; bbbb = bb*bb cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd ee = e*e ; eee = ee*e ; eeee = ee*ee v1 = ( +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd +20*bb*ccc*dd*e ) v2 = ( -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd ) return (v1+v2) == 0 t1 = ( ((1, -1, -19, -11, 30), '4 unique, real roots.'), ((4, 4,-119, -60, 675), '4 unique, real roots, B4 = 0.'), ((1, 6, -48,-182, 735), '2 equal roots.'), ((1,-12, 50, -84, 45), '2 equal roots. B4 = 0.'), ((1,-20, 146,-476, 637), '2 equal roots, 2 complex roots.'), ((1,-12, 58,-132, 117), '2 equal roots, 2 complex roots. B4 = 0.'), ((1, -2, -36, 162, -189), '3 equal roots.'), ((1,-20, 150,-500, 625), '4 equal roots. B4 = 0.'), ((1, -6, -11, 60, 100), '2 pairs of equal roots, B4 = 0.'), ((4, 4, -75,-776,-1869), '2 complex roots.'), ((1,-12, 33, 18, -208), '2 complex roots, B4 = 0.'), ((1,-20, 408,2296,18020), '4 complex roots.'), ((1,-12, 83, -282, 442), '4 complex roots, B4 = 0.'), ((1,-12, 62,-156, 169), '2 pairs of equal complex roots, B4 = 0.'), ) for (abcde, comment) in t1 : print () fourRoots = rootsOfQuartic (abcde) print (comment) print (' Coefficients =', abcde) print (' Four roots =', fourRoots) print (' Equal roots detected:', equalRoots(abcde)) # Check results. a,b,c,d,e = abcde for x in fourRoots : # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0 sum = (a*x**4 + b*x**3 + c*x**2 + d*x + e) if sum : # Create exception. 1/0 </syntaxhighlight> <syntaxhighlight> 4 unique, real roots. Coefficients = (1, -1, -19, -11, 30) Four roots = [5.0, 1.0, -2.0, -3.0] Equal roots detected: False 4 unique, real roots, B4 = 0. Coefficients = (4, 4, -119, -60, 675) Four roots = (2.5, -3.0, 4.5, -5.0) Equal roots detected: False 2 equal roots. Coefficients = (1, 6, -48, -182, 735) Four roots = [5.0, 3.0, -7.0, -7.0] Equal roots detected: True 2 equal roots. B4 = 0. Coefficients = (1, -12, 50, -84, 45) Four roots = (3.0, 3.0, 5.0, 1.0) Equal roots detected: True 2 equal roots, 2 complex roots. Coefficients = (1, -20, 146, -476, 637) Four roots = [7.0, 7.0, (3+2j), (3-2j)] Equal roots detected: True 2 equal roots, 2 complex roots. B4 = 0. Coefficients = (1, -12, 58, -132, 117) Four roots = ((3+2j), (3-2j), 3.0, 3.0) Equal roots detected: True 3 equal roots. Coefficients = (1, -2, -36, 162, -189) Four roots = [3.0, 3.0, 3.0, -7.0] Equal roots detected: True 4 equal roots. B4 = 0. Coefficients = (1, -20, 150, -500, 625) Four roots = (5.0, 5.0, 5.0, 5.0) Equal roots detected: True 2 pairs of equal roots, B4 = 0. Coefficients = (1, -6, -11, 60, 100) Four roots = (5.0, -2.0, 5.0, -2.0) Equal roots detected: True 2 complex roots. Coefficients = (4, 4, -75, -776, -1869) Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)] Equal roots detected: False 2 complex roots, B4 = 0. Coefficients = (1, -12, 33, 18, -208) Four roots = ((3+2j), (3-2j), 8.0, -2.0) Equal roots detected: False 4 complex roots. Coefficients = (1, -20, 408, 2296, 18020) Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)] Equal roots detected: False 4 complex roots, B4 = 0. Coefficients = (1, -12, 83, -282, 442) Four roots = ((3+5j), (3-5j), (3+2j), (3-2j)) Equal roots detected: False 2 pairs of equal complex roots, B4 = 0. Coefficients = (1, -12, 62, -156, 169) Four roots = ((3+2j), (3-2j), (3+2j), (3-2j)) Equal roots detected: True </syntaxhighlight> When description contains note <math>B4 = 0,</math> depressed quartic was processed as quadratic in <math>t^2.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Two Conic Sections= Examples of conic sections include: ellipse, circle, parabola and hyperbola. This section presents examples of two conic sections, circle and ellipse, and how to calculate the coordinates of the point/s of intersection, if any, of the two sections. Let one section with name <math>ABCDEF</math> have equation <math>Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.</math> Let other section with name <math>abcdef</math> have equation <math>ax^2 + by^2 + cxy + dx + ey + f = 0.</math> Because there can be as many as 4 points of intersection, a special "resolvent" quartic function is used to calculate the <math>x</math> coordinates of the point/s of intersection. Coefficients of associated "resolvent" quartic are calculated as follows: <syntaxhighlight lang=python> # python code def intersection_of_2_conic_sections (abcdef, ABCDEF) : ''' A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF) where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function: y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_ ''' A,B,C,D,E,F = ABCDEF a,b,c,d,e,f = abcdef G = ((-1)*(B)*(a) + (1)*(A)*(b)) H = ((-1)*(B)*(d) + (1)*(D)*(b)) I = ((-1)*(B)*(f) + (1)*(F)*(b)) J = ((-1)*(C)*(a) + (1)*(A)*(c)) K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c)) L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c)) M = ((-1)*(E)*(f) + (1)*(F)*(e)) g = ((-1)*(C)*(b) + (1)*(B)*(c)) h = ((-1)*(E)*(b) + (1)*(B)*(e)) A_ = (J)*(g) + (G)*(G) B_ = (J)*(h) + (K)*(g) + (2)*(G)*(H) C_ = (K)*(h) + (L)*(g) + (2)*(G)*(I) + (H)*(H) D_ = (L)*(h) + (M)*(g) + (2)*(H)*(I) E_ = (M)*(h) + (I)*(I) str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_) print (str1) return A_,B_,C_,D_,E_ </syntaxhighlight> == With no common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections06.png|thumb|400px|'''Two conic sections with no common point.''' </br> Resolvent quartic function (black curve) has no real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 80 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.98x^2 + 6850.532x - 22836.7009.</math> <math>f(x)</math> has no real roots. Therefore, there is no point of intersection. {{RoundBoxBottom}} ==With one common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections05.png|thumb|400px|'''Two conic sections with one common point.''' </br> Resolvent quartic function (black curve) has two equal, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 73 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.84x^2 + 7456.48x - 24355.36.</math> Roots of <math>f(x)</math> are: <math>(-14-22.978250586152114j),</math> <math>(-14+22.978250586152114j),</math> <math>5.8, 5.8.</math> <math>f(x)</math> has 2 equal, real roots at <math>x = 5.8,</math> effectively 1 real root where <math>x = 5.8</math> Therefore, there is one point of intersection where <math>x = 5.8.</math> {{RoundBoxBottom}} ==With two common points== ===Example 1=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections04.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two unique, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 64 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.66x^2 + 8235.556x - 26681.1841.</math> Roots of <math>f(x)</math> are: <math>(-14.361578825892241-23.341853011785357j),</math> <math>(-14.361578825892241+23.341853011785357j),</math> <math>4.59885619413921, 7.72430145764527.</math> <math>f(x)</math> has 2 unique, real roots at <math>x = 4.59885619413921, 7.72430145764527.</math> Therefore, there are two points of intersections where <math>x = 4.59885619413921, 7.72430145764527.</math> {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections01.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two pairs of equal roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 18.8x - 1.6y + 53 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 + 37.6x^3 - 504.24x^2 + 2835.04x - 5685.16.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 13, 13.</math> <math>f(x)</math> has 2 pairs of equal roots at <math>x = 5.8, 13,</math> effectively 2 real roots. Therefore, there are two points of intersection where <math>x = 5.8, 13.</math> {{RoundBoxBottom}} ==With 3 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections03.png|thumb|400px|'''Two conic sections with three common points.''' </br> Resolvent quartic function (black curve) has one pair of equal roots and 2 unique, real roots. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 17.6x - 3.2y + 55 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 6.83589838486224, 13.7641016151377 .</math> <math>f(x)</math> has 1 pair of equal roots at <math>x = 5.8</math> and 2 unique, real roots at <math>x = 6.83589838486224, 13.7641016151377,</math> effectively 3 real roots. Therefore, there are three points of intersection where <math>x = 5.8, 6.83589838486224, 13.7641016151377.</math> {{RoundBoxBottom}} ==With 4 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections02.png|thumb|400px|'''Two conic sections with four common points.''' </br> Resolvent quartic function (black curve) has 4 unique, real roots. </br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation:<math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation:<math>x^2 + y^2 - 18.8x - 1.6y + 62.99 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x)</math><math>= -x^4 + 37.6x^3 - 504.4398x^2 + 2838.79624x - 5544.61147921.</math> Roots of <math>f(x)</math> are:<math>4.36661032156283,</math><math>8.77936456353008,</math> <math>10.0206354364699,</math><math>14.4333896784371.</math> <math>f(x)</math> has 4 real roots as shown above. Therefore, there are four points of intersection where <math>x = 4.36661032156283,</math> <math>8.77936456353008,</math><math>10.0206354364699,</math><math>14.4333896784371.</math> {{RoundBoxBottom}} ===Values of y=== Section above calculates <math>x</math> coordinates of the four common points. This section calculates <math>y</math> coordinates. Here is the lazy way to do it: {{RoundBoxTop|theme=2}} [[File:0818ResQuartic.png|thumb|400px|'''Graphs of 2 curves, red and blue, with coefficients of <math>x,y</math> reversed.''' </br> Roots of "resolvent" quartic, black curve, are interpreted as values of <math>y.</math> ]] <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 # Coefficients of x,y are reversed. abcde = intersection_of_2_conic_sections ((b,a,0,e,d,f), (B,A,C,E,D,F)) values_of_y = rootsOfQuartic (abcde) print (values_of_y) </syntaxhighlight> <syntaxhighlight> [1.6215767432213488, -0.021576743221348723, 5.862095579401657, -4.262095579401658] </syntaxhighlight> By inspection of the diagram above, it is obvious that: <syntaxhighlight lang=python> # python code four_common_points = ( (4.36661032156283, 1.6215767432213488), (8.77936456353009, -4.262095579401658), (10.0206354364699, 5.862095579401657), (14.4333896784372, -0.021576743221348723), ) </syntaxhighlight> Check results: <syntaxhighlight lang=python> # python code def sum_zero (input) : ''' This function calculates sum of all values in input. If sum is very close to 0, this function returns 0. ''' sum = 0 for v in input : sum += v if sum == 0 : return sum tolerance = 1e-14 max = sorted([ abs(v) for v in input ])[-1] if abs(sum) < tolerance*max : return 0 return sum # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 for (x,y) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum1 = sum_zero(values) if sum1 : print ('Error: sum1 =',sum1) values = a*x**2, b*y**2, d*x, e*y, f sum2 = sum_zero(values) if sum2 : print ('Error: sum2 =',sum2) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. Check results with coefficients of <math>x, y</math> reversed: <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = B,A,C,E,D,F # The circle: a,b,c,d,e,f = b,a,c,e,d,f for (y,x) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum3 = sum_zero(values) if sum3 : print ('Error: sum3 =',sum3) values = a*x**2, b*y**2, d*x, e*y, f sum4 = sum_zero(values) if sum4 : print ('Error: sum4 =',sum4) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. {{RoundBoxBottom}} =Links to related topics= {{RoundBoxTop|theme=1}} [[Cubic_function#Cubic_formula | "Cubic formula"]] [[Complex_square_root | "Complex square root" ]] {{RoundBoxBottom}} [[Category:Mathematics]] kbwlpqn2vgs8ph7hm8t6972l0bkh93x 2844182 2844181 2026-11-03T11:30:02Z ThaniosAkro 2805358 /* With 4 common points */ 2844182 wikitext text/x-wiki The quartic function is the bridge between the cubic function and more advanced functions such as the quintic and sextic. =Objective= {{RoundBoxTop|theme=3}} [[file:Books-aj.svg_aj_ashton_01f.png|right|100px]] * Present quartic function and quartic equation. * Introduce the concept of roots of equal absolute value. * Show how to predict and calculate equal roots, techniques that will be useful when applied to higher order functions. * Simplify the depressed quartic. * Show that the quartic equation is effectively solved when at least one root is known. * Present the "resolvent" cubic function. * Show how to derive and use the quartic formula. {{RoundBoxBottom}} =Lesson= {{RoundBoxTop|theme=2}} ==Introduction== The quartic function is the sum of powers of <math>x</math> from <math>0</math> through <math>4</math>: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx^1 + ex^0</math> usually written as: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx + e.</math> If <math>e == 0</math> the function becomes <math>x(ax^3 + bx^2 + cx + d).</math> Within this page we'll say that: * both coefficients <math>a, e</math> must be non-zero, * coefficient <math>a</math> must be positive (simply for our convenience), * all coefficients must be real numbers, accepting that the function may contain complex roots. The quartic equation is the quartic function equated to zero: <math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>. Roots of the function are values of <math>x</math> that satisfy the quartic equation. {{RoundBoxTop|theme=1}} Because the function is "quartic" (maximum power of <math>x</math> is <math>4</math>), the function contains exactly <math>4</math> roots, an even number of complex roots and an even number of real roots. Other combinations of real and complex roots are possible, but they produce complex coefficients. {{RoundBoxBottom}} {{RoundBoxTop|theme=7}} [[File:0312quartic01a.png|thumb|400px|'''Graph of typical quartic function showing minima and maximum.''']] The figure shows a typical quartic function. The function crosses the <math>X</math> axis in 4 different places. The function has 4 roots: <math>(-2,0), (1,0), (5,0), (10,0).</math> This function contains one local minimum, one local maximum and one absolute minimum. There is no absolute maximum. Because the function contains one absolute minimum: * If abs(<math>x</math>) is very large, <math>f(x)</math> is always positive. * If absolute minimum is above <math>X</math> axis, curve does not cross <math>X</math> axis and function contains only complex roots. * There is always at least one point where the curve is parallel to <math>X</math> axis. The curve is never parallel to the <math>Y</math> axis. For any real value of <math>x</math> there is always a real value of <math>y.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0312quartic02.png|thumb|400px|'''When coefficient d is missing, there is a stationary point at x = 0.''']] If coefficient <math>d</math> is missing, the quartic function becomes <math>y = ax^4 + bx^3 + cx^2 + e,</math> and <math>y' = 4ax^3 + 3bx^2 + 2cx = x(4ax^2 + 3bx + 2c).</math> For a stationary point <math>y' = x(4ax^2 + 3bx + 2c) = 0.</math> When coefficient <math>d</math> is missing, there is always a stationary point at <math>x = 0.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} [[File:0319quartic02.png|thumb|400px|'''Graph of quartic function that is quadratic in <math>x^2</math>.'''</br>Because coefficient <math>d</math> is missing, there is a stationary point where <math>x = 0.</math>]] If coefficients <math>b, d</math> are missing, the quartic function becomes a quadratic in <math>x^2.</math> The curve (red line) in diagram has equation: <math>y = f(x) = \frac{x^4 - 13x^2 + 36}{5}</math> The quartic equation may be solved as: <math>X^2 - 13X + 36 = 0</math> where <math>X = x^2</math> or <math>x = \sqrt{X}.</math> <math>X = 4</math> or <math>X = 9.</math> <math>x = \pm 2</math> or <math>x = \pm 3.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0312quartic03.png|thumb|400px| ]] The quartic function may be expressed as <math>x = ay^4 + by^3 + cy^2 + dy + e.</math> Unless otherwise noted, references to "quartic function" on this page refer to function of form <math>y = ax^4 + bx^3 + cx^2 + dx + e.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0318quartic01.png|thumb|400px|'''Graph of quartic function with coefficient a negative.'''</br>There is no absolute minimum.]] Coefficient <math>a</math> may be negative as shown in diagram. As <code>abs</code><math>(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^4.</math> When <code>abs</code><math>(x)</math> is very large, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "quartic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0502quartic01.png|thumb|400px|'''Graph of quartic function with coefficient b missing.''' </br> Sum of roots is <math>0.</math> </br> <math>Y</math>axis compressed for clarity. ]] When sum of roots is <math>0,</math> coefficient <math>b = 0.</math> In the diagram, roots of <math>f(x)</math> are <math>-5, -4, 2, 7.</math> Sum of roots <math>= 0.</math> Therefore coefficient <math>b = 0.</math> {{RoundBoxBottom}} ==Function as product of linear function and cubic== {{RoundBoxTop|theme=3}} [[File:0313quartic01.png|thumb|400px|'''Graphs of quartic function and associated cubic function.''']] When <math>p</math> is a root of the function, the function may be expressed as: <math>(x - p)( Ax^3 + Bx^2 + Cx + D )</math> where <math>A = a;\ B = Ap + b;\ C = Bp + c;\ D = Cp + d.</math> When one real root <math>p</math> is known, the other three roots may be calculated as roots of the cubic function <math>Ax^3 + Bx^2 + Cx + D.</math> In the diagram the quartic function has equation: <math>y = \frac{ x^4 - 23x^3 + 163x^2 - 393x + 252 }{ 48 }.</math> It is known that <math>3</math> is a root of this function. The associated cubic has equation: <math>y = \frac { x^3 - 20x^2 + 103x - 84} { 48 }</math> The 2 curves coincide at points <math>(1, 0),\ (7, 0),\ (12, 0),</math> the three points that are roots of both functions. <math></math> {{RoundBoxBottom}} ==Function defined by 5 points== {{RoundBoxTop|theme=2}} [[File:0313quartic02.png|thumb|400px|'''Figure 1. Quartic function defined by 5 points''' </br>Any 5 points on the curve may be used to define the function.]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 5 unique points on the curve. For example, let us choose the five points: <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code points = (-5,0), (-2,0), (1,0), (3,-6), (6,2) L11 = [] for point in points : x,y = point L11 += [[x*x*x*x, x*x*x, x*x, x, 1, -y]] print (L11) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ 625.0, -125.0, 25.0, -5.0, 1.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 0.0], # [ 1.0, 1.0, 1.0, 1.0, 1.0, 0.0], # matrix supplied to function solveMbyN() below. [ 81.0, 27.0, 9.0, 3.0, 1.0, 6.0], # 5 rows by 6 columns. [1296.0, 216.0, 36.0, 6.0, 1.0, -2.0]] # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L11) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.02651515151515152, 0.004545454545454542, -0.847727272727273, -0.728787878787879, 1.5454545454545459) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by the 5 points <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> is <math>y = \frac{0.875 x^4 + 0.15 x^3 - 27.975 x^2 - 24.05 x + 51}{33}.</math> {{RoundBoxBottom}} ==Function defined by 3 points and 2 slopes== {{RoundBoxTop|theme=2}} [[File:0314quartic03.png|thumb|400px|'''Figure 1. Quartic function defined by 3 points and 2 slopes.''' </br>Any 3 points on the curve and the slopes at any 2 of these points may be used to define the function.</br></br>Slope at <math>(-2, -2)\ =</math> slope at <math>(6, -4)\ =\ 0.</math>]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 3 unique points on the curve and the slopes at any 2 of these points. For example, let us choose the three points: <math>(-2, -2), (6, -4), (4, 1)</math> At point <math>(-2, -2)</math> slope is <math>0.</math> At point <math>(6, -4)</math> slope is <math>0.</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> Rearrange the standard cubic function of slope to prepare for the calculation of <math>a,b,c,d,e:</math> <math> 4 x^3 a + 3 x^2 b + 2 x c + 1 d + 0 e - s = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code def makeEntry(input) : x,y,s = ( tuple(input) + (None,) )[:3] L1 = [] if s != None : L2 = [ float(v) for v in [4*x*x*x, 3*x*x, 2*x, 1, 0, -s] ] L1 += [ L2 ] L2 = [ float(v) for v in [x*x*x*x, x*x*x, x*x, x, 1, -y]] L1 += [ L2 ] return L1 t1 = ( (-2,-2, 0), # point (-2, -2) with slope 0. (6,-4, 0), # point (6, -4) with slope 0. (4,1), # point (4, 1) ) L1 = [] for v in t1 : L1 += makeEntry ( v ) print (L1) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ -32.0, 12.0, -4.0, 1.0, 0.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 2.0], # [ 864.0, 108.0, 12.0, 1.0, 0.0, 0.0], # matrix supplied to function solveMbyN() below. [1296.0, 216.0, 36.0, 6.0, 1.0, 4.0], # 5 rows by 6 columns. [ 256.0, 64.0, 16.0, 4.0, 1.0, -1.0]]. # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L1) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.03255208333333339, -0.2526041666666665, -0.3072916666666667, 2.84375, 2.375) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by three points and two slopes is: <math>y = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> ===Associated cubic functions=== {{RoundBoxTop|theme=3}} ====When p == -2==== {{RoundBoxTop|theme=1}} [[File:0314_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = -2.''' </br>In this case roots of associated cubic include x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == -2,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 15.25 x^2 + 15.75 x + 105 } {48}.</math> Three blue vertical lines show 3 values of <math>x</math> where <math>g(x) = 0</math> and <math>f(x) = f(-2)</math> In this case roots of <math>g(x)</math> include <math>x = p.</math> {{RoundBoxBottom}} ====When p == 5==== {{RoundBoxTop|theme=1}} [[File:0314_2curves02.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 5.''' </br>In this case the one root of associated cubic excludes x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 5,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 4.3125 x^2 - 36.3125 x - 45.0625 } {48}.</math> Two blue vertical lines show 2 values of <math>x</math> where <math>f(x) = f(5)</math> In this case the one root of <math>g(x)</math> excludes <math>x = p.</math> {{RoundBoxBottom}} ====When p == 6==== {{RoundBoxTop|theme=1}} [[File:0315_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 6.''' </br>In this case the one root of associated cubic includes x = p.]] Quartic function is: <math>y = f(x) </math><math>= \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 6,</math> associated cubic function is: <math>y = g(x) </math><math>= \frac{ 1.5625 x^3 - 2.75 x^2 - 31.25 x - 51 } {48}.</math> One blue vertical line shows 1 value of <math>x</math> where <math>f(x) = f(6)</math> In this case the one root of <math>g(x)</math> includes <math>x = p.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} ==Examples== {{RoundBoxTop|theme=2}} ===Quartic with 2 stationary points=== {{RoundBoxTop|theme=1}} [[File:0315_2statPoints02.png|thumb|400px|'''Graph of quartic function with 2 stationary points.''']] In the diagram the red line represents quartic function <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 2 roots of <math>g(x),\ -1</math> and <math>\frac{-1}{4}</math> show that <math>f(x)</math> has stationary points at <math>x = -1</math> and <math>x = -0.25.</math> {{RoundBoxBottom}} ===Quartic with 1 stationary point=== {{RoundBoxTop|theme=1}} [[File:0315_1statPoint.png|thumb|400px|'''Graph of quartic function with 1 stationary point.''']] In the diagram the red line represents quartic function <math>y = f(x) = \frac{x^4 + 3x^3 + 3x^2 + 3x}{2}</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 1 root of <math>g(x),\ -1.607</math> (approx.), shows that <math>f(x)</math> has 1 stationary point where <math>g(x) = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =First and second derivatives= {{RoundBoxTop|theme=1}} ==Points of inflection== {{RoundBoxTop|theme=2}} [[File:0317_3curves01.png|thumb|400px|'''Graphs of quartic function and first two derivatives.'''</br>Dotted portion of black line shows where f(x) is always concave down.</br>Dotted portion of red line shows where g(x) is decreasing.</br>Dotted portion of blue line shows where h(x) is negative.</br>When h(x) is negative, f(x) is concave down.</br>When h(x) is positive, f(x) is concave up.]] In the diagram the black line has equation: <math>y = f(x) = \frac{x^4 - 16x^3 + 42x^2 + 12x + 4}{144}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{x^3 - 12x^2 + 21x + 3}{36}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{x^2 - 8x + 7}{12}.</math> When <math>x < x_1:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. When <math>x == x_1:</math> * <math>y'</math> is at a local maximum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between up and down. When <math>x_1 < x < x_2:</math> * <math>y'</math> is decreasing. * <math>y''</math> is negative. * <math>f(x)</math> is always concave down. When <math>x == x_2:</math> * <math>y'</math> is at a local minimum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between down and up. When <math>x_2 < x:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. {{RoundBoxTop|theme=1}} The roots of <math>h(x): x_1 = 1, x_2 = 7.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At point <math>p_1</math> concavity of <math>f(x)</math> changes from up to down. At point <math>p_2</math> concavity of <math>f(x)</math> changes from down to up. The points <math>p_1, p_2</math> (the <math>X</math> coordinates of which are roots of <math>h(x)</math>) are the '''points of inflection''' of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Maxima and minima== {{RoundBoxTop|theme=2}} [[File:0317_3curves03.png|thumb|400px|'''Graphs of quartic function and first two derivatives showing maximum and minima.'''</br> Point <math>p_{2}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{2}</math> is concave down. Point <math>p_{2}</math> is local maximum. </br> Point <math>p_{1}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{1}</math> is concave up. Point <math>p_{1}</math> is local minimum. </br> Similarly, point <math>p_{3}</math> is local minimum. ]] In the diagram the black line has equation: <math>y = f(x) = \frac{1.5625x^4 - 12.125x^3 - 14.75x^2 + 136.5x + 114}{48}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{6.25x^3 - 36.375x^2 - 29.5x + 136.5}{48}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{18.75x^2 - 72.75x - 29.5}{48}.</math> Roots of <math>g(x):\ x_1 = -2;\ x_2 = 1.82;\ x_3 = 6.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> At <math>x_1\ h(x_1)</math> is positive. Point <math>p_1</math> is a stationary point and <math>f(x)</math> at <math>p_1</math> is concave up. Point <math>p_1</math> is a local minimum. Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At <math>x_2\ h(x_2)</math> is negative. Point <math>p_2</math> is a stationary point and <math>f(x)</math> at <math>p_2</math> is concave down. Point <math>p_2</math> is a local maximum. Let point <math>p_3</math> on <math>f(x)</math> have coordinates <math>(x_3, f(x_3)).</math> At <math>x_3\ h(x_3)</math> is positive. Point <math>p_3</math> is a stationary point and <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is a local minimum. {{RoundBoxBottom}} ==Quartic with 2 stationary points== {{RoundBoxTop|theme=3}} [[File:0318_3curves01.png|thumb|400px|'''Graph of quartic function with 2 stationary points and first 2 derivatives.''' </br> Black line: <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> </br> Red line: <math>y' = g(x) = 4(4x^3 + 9x^2 + 6x + 1).</math> </br> <math>y'' = h(x) = 4(12x^2 + 18x + 6)</math> </br> Blue line: <math>y'' = 4(2x^2 + 3x + 1) = \frac{h(x)}{6}.</math> </br> Dotted portion of black line shows where <math>f(x)</math> is concave down. ]] In the diagram, point <math>p_1</math> on <math>f(x)</math> has coordinates <math>(x_1, f(x_1)).</math> Similarly, points <math>p_2, p_3</math> have coordinates <math>(x_2, f(x_2)),\ (x_3, f(x_3)).</math> <math>y'</math> has roots: <math>x_1 = -1;\ x_3 = -0.25.</math> Points <math>p_1, p_3</math> are stationary points. <math>y''</math> has roots: <math>x_1 = -1;\ x_2 = -0.5.</math> Points <math>p_1, p_2</math> are points of inflection. At point <math>p_3\ y''</math> is positive. <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is local minimum. {{RoundBoxTop|theme=2}} Summary: * Point <math>p_1</math> is both stationary point and point of inflection. * Point <math>p_2</math> is point of inflection. * Point <math>p_3</math> is both stationary point and local minimum. {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =The simplest quartic function= {{RoundBoxTop|theme=2}} [[File:0320quartic01.png|thumb|400px|'''Graph of simplest quartic function.''' </br> Point <math>p_1\ (-1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_2\ (1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_0\ (0, -1.1^4)</math> is <math>Y</math> intercept of <math>f(x).</math> ]] The simplest quartic function has coefficients <math>b = c = d = 0.</math> Red line in diagram has equation: <math>y = f(x) = x^4 - 1.1^4</math> First derivative (not shown) of <math>f(x):\ y' = g(x) = 4x^3.</math> When <math>x == 0,\ g(x) = 0.</math> There is a stationary point on <math>f(x)</math> when <math>x == 0,</math> point <math>p_0.</math> Second derivative (not shown) of <math>f(x):\ y'' = h(x) = 12x^2.</math> When <math>x == 0,\ h(x) = 0.</math> There is a point of inflection on <math>f(x)</math> when <math>x == 0.</math> For every non-zero value of <math>x,\ h(x)</math> is positive. To left and right of point <math>p_0,\ f(x)</math> is always concave up. Point <math>p_0</math> is both local minimum and absolute minimum. * Point <math>p_0</math> is stationary point and point of inflection and absolute minimum. Curve <math>f(x)</math> is useful for finding the fourth root of a real number. Solve: <math>x = N^{\frac{1}{4}}.</math> <math>x^4 = N.</math> <math>x^4 - N = 0.</math> This is equivalent to finding a root of function <math>y = j(x) = x^4 - N.</math> If you use Newton's method to find a root of <math>j(x),</math> this would be more efficient than solving <math>x = \sqrt{\sqrt{N}}.</math> {{RoundBoxBottom}} =Roots of equal absolute value= {{RoundBoxTop|theme=1}} The standard quartic function: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> For <math>x</math> in <math>(1)</math> substitute <math>(p+q).</math> Call this <math>(2).</math> For <math>x</math> in <math>(1)</math> substitute <math>(p-q).</math> Call this <math>(3).</math> Combine <math>(2)</math> and <math>(3)</math> to eliminate <math>q</math> and produce an equation in <math>p:</math> <math>(- 64aaa)pppppp+</math> <math>(- 96aab)ppppp+</math> <math>(- 32aac - 48abb)pppp+</math> <math>(- 32abc - 8bbb)ppp+</math> <math>(+ 16aae - 4abd - 4acc - 8bbc)pp+</math> <math>(+ 8abe - 2bbd - 2bcc)p+</math> <math>(+ add + bbe - bcd)\ =\ 0\ \dots\ (4).</math> We are interested in coefficient <math>0</math> of <math>(4):\ c_0 = add + bbe - bcd.</math> If <math>c_0 == 0,\ p=0</math> is a solution and function <math>(1)</math> has 2 roots of form <math>0 \pm q</math> where <math>q = \sqrt{\frac{-d}{b}}.</math> An example: {{RoundBoxTop|theme=3}} [[File:0320quartic02.png|thumb|400px|'''Graph of quartic function with 2 roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 12x^3 + 31x^2 + 48x -140}{45}.</math> <math>a,b,c,d,e = 1,-12,31,48,-140</math> <math>c_0 = add + bbe - bcd </math><math>= 1(48)(48) + (-12)(-12)(-140) - (-12)(31)(48) = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-48}{-12}} = \sqrt{4} = \pm 2.</math> The 2 roots of equal absolute value are: <math>2, -2.</math> {{RoundBoxBottom}} The method works with complex roots of equal absolute value: {{RoundBoxTop|theme=2}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with 2 complex roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}.</math> <math>a,b,c,d,e = 1,-3,-1,-27,-90</math> <math>c_0 = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-(-27)}{-3}} = \sqrt{-9} = \pm 3i.</math> The 2 roots of equal absolute value are: <math>3i, -3i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Equal roots= {{RoundBoxTop|theme=2}} Equal roots occur when the function and the slope of the function both equal zero. <math>ax^4 + bx^3 + cx^2 + dx + e = 0\ \dots\ (1)</math> <math>4ax^3 + 3bx^2 + 2cx + d = 0\ \dots\ (2)</math> Begin the process of reducing <math>(1),\ (2)</math> to linear functions. Combine <math>(1),\ (2)</math> to produce 2 cubic functions: <math>Fx^3 + Gx^2 + Hx + J\ \dots\ (1a)</math> where: <math>F = ad;\ G = bd - 4ae;\ H = cd - 3be;\ J = dd - 2ce.</math> <math>fx^3 + gx^2 + hx + j\ \dots\ (2a)</math> where: <math>f = 4a;\ g = 3b;\ h = 2c;\ j = d.</math> Combine <math>(1a),\ (2a)</math> to produce 2 quadratic functions: <math>Kx^2 + Lx + M\ \dots\ (1b)</math> where: <math>K = Gf -Fg;\ L = Hf -Fh;\ M = Jf - Fj.</math> <math>kx^2 + lx + m\ \dots\ (2b)</math> where: <math>k = Fj - Jf;\ l = Gj - Jg;\ m = Hj - Jh.</math> Combine <math>(1b),\ (2b)</math> to produce 2 linear functions: <math>Rx + S\ \dots\ (1c)</math> where: <math>R = Lk - Kl;\ S = Mk - Km.</math> <math>rx + s\ \dots\ (2c)</math> where: <math>r = Km-Mk;\ s = Lm - Ml.</math> From <math>(1c):\ x_1 = \frac{-S}{R}</math> From <math>(2c):\ x_2 = \frac{-s}{r}</math> If <math>x_1 == x_2:</math> <math>\frac{-S}{R} = \frac{-s}{r}</math> <math>Rs = rS</math> <math>Rs - Sr = 0.</math> The value <math>Rs - Sr</math> is in fact: <syntaxhighlight> + 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee - 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd + 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee + 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde + 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee - 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde - 3aabbccdddddd </syntaxhighlight> which, by removing values <math>aa, d</math><math>d</math> (common to all values), may be reduced to: <syntaxhighlight> status = ( + 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee - 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd + 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee + 384abccddde - 54abcddddd + 128acccccee - 80accccdde + 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee - 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde - 3bbccdddd ) </syntaxhighlight> If <math>status == 0,</math> there are at least 2 equal roots which may be calculated as shown below. {{RoundBoxTop|theme=2}} If coefficient <math>d</math> is non-zero, it is not necessary to calculate <math>status.</math> If coefficient <math>d == 0,</math> verify that <math>status = 0</math> before proceeding. {{RoundBoxBottom}} ==No equal roots== {{RoundBoxTop|theme=1}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with no equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}</math> <math>a,b,c,d,e = 1, -3, -1, -27, -90</math> <math>R, S = 15269148, -35977608</math> <math>x_1 = \frac{-S}{R} = \frac{35977608}{15269148} = 2.3562289133617\dots</math> <math>r, s = 35977608, -60634332</math> <math>x_2 = \frac{-s}{r} = \frac{60634332}{35977608} = 1.685335278543253\dots</math> <math>x_1 != x_2.</math> There are no equal roots. {{RoundBoxBottom}} ==Exactly 2 equal roots== {{RoundBoxTop|theme=1}} [[File:0329quartic04.png|thumb|400px|'''Graph of quartic function with exactly 2 equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 + 6x^3 - 48x^2 - 182x + 735}{100}</math> <math>a,b,c,d,e = 1, 6, -48, -182, 735</math> <math>R, S = -1027353600, -7191475200</math> <math>x_1 = \frac{-S}{R} = \frac{7191475200}{-1027353600} = -7</math> <math>r, s = 7191475200, 50340326400</math> <math>x_2 = \frac{-s}{r} = \frac{-50340326400}{7191475200} = -7</math> <math>x_1 = x_2 = -7.</math> There are 2 equal roots at <math>x = -7.</math> {{RoundBoxTop|theme=2}} The following 3 graphs show the steps that lead to calculation of equal roots at point <math>(-7, 0).</math> <gallery> File:0914quartic00.png|<small>Figure 1. graph of <math>f(x)</math> and 2 associated cubic functions.</small> File:0914quartic01.png|<small>Figure 2. graph of <math>f(x)</math> and 2 associated quadratic functions.</small> File:0914quartic02.png|<small>Figure 3. graph of <math>f(x)</math> and 2 associated linear functions.</small> </gallery> In all graphs, all curves have a common root at point <math>(-7, 0).</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=4}} See [[Quartic_function#Function_as_product_of_linear_function_and_cubic | "Function as product of linear function and cubic"]] above. To calculate all roots: <syntaxhighlight lang=python> # python code. a,b,c,d,e = 1,6,-48,-182,735 # The associated cubic: p = -7 A = a B = A*p + b C = B*p + c D = C*P + d # The associated quadratic: a1 = A b1 = a1*p + B c1 = b1*p + C a1,b1,c1 </syntaxhighlight> <syntaxhighlight> (1, -8, 15) </syntaxhighlight> Roots of quadratic function <math>g(x) = x^2 - 8x + 15</math> are <math>3, 5.</math> All roots of <math>f(x)</math> are <math>-7, -7, 3, 5.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Exactly 3 equal roots== {{RoundBoxTop|theme=1}} [[File:0320_2curves01.png|thumb|400px|'''Graph of quartic function with exactly 3 equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 2x^3 - 36x^2 + 162x - 189}{100}</math> <math>a,b,c,d,e = 1, -2, -36, 162, -189</math> <math>R, S = 0, 0\ \dots\dots\ (1c)</math> <math>r, s = 0, 0\ \dots\dots\ (2c)</math> In this case the calculation of <math>x_1, x_2</math> is not appropriate because there are more than 2 equal roots. Try equations <math>(1b), (2b).</math> Both of these are equivalent to: <math>y = g(x) = x^2 - 6x + 9,</math> blue line in diagram. Discriminant of <math>g(x) = (-6)^2 - 4(1)(9) = 0.\ g(x)</math> has two equal roots at <math>x = \frac{-(-6)}{2(1)} = 3.</math> Therefore <math>f(x)</math> has 3 equal roots at <math>x = 3.</math> {{RoundBoxBottom}} ==Four equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves01.png|thumb|400px|'''Graph of quartic function with 4 equal roots and corresponding cubic.''' </br> <math>g(x) = \frac{f'(x)}{4}.</math> ]] Red line in diagram is of function: <math>y = f(x) = x^4 - 20x^3 + 150x^2 - 500x + 625.</math> <math>a,b,c,d,e = 1, -20, 150, -500, 625</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> <math>K, L, M = 0, 0, 0</math> <math>k, l, m = 0, 0, 0</math> In this case <math>(1b), (2b), (1c), (2c)</math> are all null. This is the only case in which <math>(1b), (2b)</math> are null. <math>(1a), (2a)</math> are both equivalent to: <math>y = g(x) = x^3 - 15x^2 + 75x - 125,</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 4 equal roots at <math>x = 5.</math> {{RoundBoxBottom}} ==Two pairs of equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 6x^3 - 11x^2 + 60x + 100}{20}.</math> <math>a,b,c,d,e = 1, -6, -11, 60, 100</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> In this case <math>(1c), (2c)</math> are both null. <math>(1b), (2b)</math> are both equivalent to: <math>y = g(x) = \frac{x^2 - 3x - 10}{20},</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = -2</math> and one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 2 equal roots at <math>x = -2</math> and 2 equal roots at <math>x = 5.</math> {{RoundBoxTop|theme=2}} [[File:0408_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal, complex roots and corresponding quadratic.''' </br> <math>g(x) = (x-(3-2i))(x-(3+2i)) = x^2 - 6x + 13.</math> </br> <math>f(x) = (g(x))^2.</math> ]] This method is valid for complex roots. For example: <math>y = f(x) = x^4 - 12x^3 + 62x^2 - 156x + 169.</math> <math>a,b,c,d,e = 1,-12,62,-156,169.</math> In this case <math>(1c),\ (2c)</math> are both null. <math>(1b),\ (2b)</math> are both equivalent to: <math>y = g(x) = x^2 - 6x + 13,</math> blue line in diagram. Roots of <math>g(x)</math> are <math>3 \pm 2i.</math> <math>f(x)</math> has 2 roots equal to <math>3 + 2i</math> and 2 roots equal to <math>3 - 2i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Summmary== {{RoundBoxTop|theme=1}} {| class="wikitable" |- ! || No equal roots !! 2 equal roots !! 3 equal roots !! 4 equal roots !! 2 pairs of equal roots |- | Cubic: 1(a), 2(a) | different | different | different | same | different |- | Quadratic: 1(b), 2(b) | different | different | same, 1root | null | same, 2roots |- | Linear: 1(c), 2(c) | different | same | null | null | null |} {{RoundBoxBottom}} ==Caution== {{RoundBoxTop|theme=4}} [[File:0915quartic02.png|thumb|400px| Calculation of false equal roots.</br> In this example, method calculates 2 legitimate equal roots at <math>(4, 0)</math> and 2 false equal roots at <math>(7.543296089385474,0).</math> <math></math> ]] Black line in diagram has equation: <math>y = f(x) = 0.012684240362811794x^4</math> <math> -\ 0.19522392290249435x^3 </math> <math> +\ 0.7654478458049887x^2 + 0x - 3. </math> <math>f(x)</math> is a quartic function with exactly 2 equal roots and coefficient <math>d</math> missing. Calculation of equal roots of <math>f(x)</math> gives linear functions <math>(1c), (2c)</math> null and quadratic functions <math>(1b), (2b)</math> with equal roots of <math> (4,0), (7.543296089385474,0). </math> Usually, this indicates that <math>f(x)</math> should have 2 equal roots at <math>(4,0)</math> and 2 equal roots at <math>(7.543296089385474,0).</math> It is obvious that <math>7.543296089385474</math> is not a root of <math>f(x).</math> When <math>x = 7.543296089385474,</math> slope of derivative <math>g(x) = 0.</math> Value of <math>f(x)\ != 0.</math> This example indicates that it would be wise to verify that calculated equal roots are in fact valid roots of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Depressed quartic= {{RoundBoxTop|theme=1}} A depressed quartic is any quartic function with any one or more of coefficients <math>b,c,d</math> missing. Within this section a depressed quartic has coefficient <math>b</math> missing. To produce the depressed quartic: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> <math>y = \frac{(4^4 a^3)(ax^4 + bx^3 + cx^2 + dx + e)}{4^4 a^3}\ \dots\ (2)</math> Let <math>x = \frac{-b + t}{4a}.</math> Substitute in <math>(2),</math> expand and simplify: <math>y = \frac{t^4 + At^2 + Bt + C}{4^4a^3}\ \dots\ (3)</math> where: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> When equated to <math>0,\ (3)</math> becomes the depressed equation: <math>t^4 + At^2 + Bt + C = 0\ \dots\ (4).</math> Be prepared for the possibility that any 1 or more of <math>A, B, C</math> may be zero. ==Coefficient B missing== {{RoundBoxTop|theme=2}} [[File:0322_3curves01.png|thumb|400px|'''Graph of quartic function that resembles a quadratic.''' ]] If coefficient <math>B == 0,\ (4)</math> becomes a quadratic in <math>t^2:</math> <math>t^4 + At^2 + C = 0.</math> <math>(1)</math> has the appearance of a quadratic. The black line: <math>y = f(x) = \frac{x^4 - 4x^3 + 9x^2 - 10x + 5}{10}</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>= 8( 8(-10) - 4(-4)(9) + -64 )</math> <math>= 8(-80 + 144 - 64)</math> <math>= 8(0) = 0.</math> The red line: <math>y' = g(x) = \frac{4x^3 - 12x^2 + 18x - 10}{10}</math> <math>y' = g(x) = 0</math> where <math>x = 1.</math> The grey line: <math>y'' = h(x) = \frac{12x^2 - 24x + 18}{10}</math> * Absolute minima of <math>f(x)</math> and of <math>h(x)</math> and point of inflection of <math>g(x)</math> occur where <math>x = \frac{-b}{4} = 1.</math> * <math>y''</math> is always positive. <math>f(x)</math> is always concave up. * <math>f(1 + p) = f(1-p).</math> {{RoundBoxTop|theme=3}} If <math>(1)</math> contains 2 pairs of equal roots, coefficient <math>B = 0.</math> The converse is not necessarily true. If <math>(1)</math> contains 4 equal roots, coefficients <math>A = B = C = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Coefficient C missing== {{RoundBoxTop|theme=2}} [[File:0501quartic01.png|thumb|400px|'''Graph of quartic function with coefficient C of depressed function missing.''' </br> <math>Y</math> axis compressed for clarity. ]] If coefficient <math>C == 0,\ (4)</math> becomes: <math>t^4 + At^2 + Bt = t(t^3 + At + B) = 0</math> in which case <math>t = 0</math> is a solution and <math>x = \frac{-b}{4a}</math> is a root of <math>(1).</math> Curve (red line) in example has equation: <math>y = f(x) = 8x^4 + 16x^3 + 24x^2 + 89x + 40.</math> Coefficients of depressed function are: <syntaxhighlight lang=python> # python code a,b,c,d,e = 8, 16, 24, 89, 40 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b A,B,C </syntaxhighlight> <syntaxhighlight> (1536, 299008, 0) </syntaxhighlight> Coefficient <math>C</math> of depressed function is missing. <math>t = 0</math> is a solution. Using <math>x = \frac{-b + t}{4a},</math> one root of <math>f(x) = \frac{-16 + 0}{4(8)} = -0.5.</math> {{RoundBoxBottom}} ==Resolvent cubic== {{RoundBoxTop|theme=1}} This section introduces a special cubic function called "resolvent" because it helps to resolve a requirement, the calculation of the roots of the quartic. The depressed quartic: <math>t^4 + At^2 + Bt + C\ \dots\ (1)</math> For <math>t</math> substitute <math>(u+v)\ \dots\ (2)</math> For <math>t</math> substitute <math>(u-v)\ \dots\ (3)</math> <math>(2)+(3):\ 2Auu + 2Avv + 2Bu + 2C + 2uuuu + 12uuvv + 2vvvv\ \dots\ (4)</math> Simplify <math>(4):\ Auu + AV + Bu + C + uuuu + 6uuV + VV\ \dots\ (4a)</math> <math>(2)-(3):\ 4Auv + 2Bv + 8uuuv + 8uvvv\ \dots\ (5)</math> Simplify <math>(5):\ 2Au + B + 4uuu + 4uV\ \dots\ (5a)</math> From <math>(5a):\ 4uV = -(2Au + B + 4uuu)\ \dots\ (5b)</math> <math>(4a)*4u4u:\ 4u4uAuu + A4u(4uV) + 4u4uBu + 4u4uC + 4u4uuuuu + 6uu4u(4uV) + (4uV)(4uV)\ \dots\ (6)</math> In <math>(6)</math> replace <math>4uV</math> with <math>(-(2Au + B + 4uuu)),</math> expand, simplify, gather like terms and result is: <math>Pu^6 + Qu^4 + Ru^2 + S</math> or {{RoundBoxTop|theme=2}} <math>PU^3 + QU^2 + RU + S\ \dots\ (7)</math> where: <math>U = u^2</math> <math>P = 64</math> <math>Q = 32A</math> <math>R = 4A^2 - 16C</math> <math>S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} From <math>(5b):\ V = v^2 = \frac{-(2Au + B + 4uuu)}{4u} = -(\frac{A}{2} + U) - \frac{B}{4u}\ \dots\ (8)</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Some simple changes reduce the number of calculations and also the sizes of coefficients <math>P, Q, R, S.</math> <math>A2 = 8ac - 3b^2</math> where <math>A2 = \frac{A}{2}</math> <math>B4 = 16a^2 d - 8abc + 2b^3</math> where <math>B4 = \frac{B}{4}</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> Then: <math>P = 64</math> <math>Q = 32(A2)(2) = 64A2</math> <math>R = 4(A2)(2)(A2)(2) - 16C = 16A2^2 - 16C</math> <math>S = -(B4)(4)(B4)(4) = -16 B4^2</math> Divide all 4 coefficients by <math>16:</math> <math>P = 4</math> <math>Q = 4A2</math> <math>R = A2^2 - C</math> <math>S = -B4^2</math> <math>V = v^2 = -(A2 + U) - \frac{B4}{u}.</math> A close examination of coefficients <math>R, S</math> shows that both coefficients are always exactly divisible by <math>4.</math> Therefore, all four coefficients may be defined as follows: <math>P = 1</math> <math>Q = A2</math> <math>R = \frac{A2^2 - C}{4}</math> <math>S = \frac{-B4^2}{4}</math> Actual calculations of values <math>R, S</math> are: <syntaxhighlight lang=python> # python code. R = ( (16)*(a)*(a)*(b)*(d) + (16)*(a)*(a)*(c)*(c) + (3)*(b)*(b)*(b)*(b) + (-16)*(a)*(b)*(b)*(c) + (-64)*(a)*(a)*(a)*(e) ) S = ( (64)*(a)*(a)*(a)*(b)*(c)*(d) + (8)*(a)*(b)*(b)*(b)*(b)*(c) + (-16)*(a)*(a)*(b)*(b)*(b)*(d) + (-16)*(a)*(a)*(b)*(b)*(c)*(c) + (-1)*(b)*(b)*(b)*(b)*(b)*(b) + (-64)*(a)*(a)*(a)*(a)*(d)*(d) ) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =Solving quartic equation= This section presents 4 examples that show how to use the depressed quartic and the resolvent cubic to solve the quartic equation. ==Four real roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves01.png|thumb|400px|'''Graphs of quartic function with 4 real roots and associated resolvent cubic.''' </br>Resolvent cubic contains 3 real, positive roots. ]] Calculate roots of: <math>y = f(x) = x^4 - x^3 - 19x^2 - 11x + 30</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,-1,-19,-11,30 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -310 -1320 6669 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 -9920 277696 -1742400 </syntaxhighlight> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = 64x^3 - 9920x^2 + 277696x - 1742400.</math> There are 3 real, positive roots: <math>9, 25, 121.</math> Using 3 roots of <math>g(x),</math> calculate 4 roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code for U in (9, 25, 121) : print ('\nU =', U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** .5 for t in (u+v, u-v) : x = (-b+t) / (4*a) y = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x:',x, '; y:',y) </syntaxhighlight> <syntaxhighlight> U = 9 x: 5.0 ; y: 0.0 x: -3.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 U = 25 x: 5.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -3.0 ; y: 0.0 U = 121 x: 5.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: -3.0 ; y: 0.0 </syntaxhighlight> Roots of <math>f(x)</math> are: <math>5, 1, -2, -3.</math> {{RoundBoxBottom}} All 3 values of <math>U</math> produce the same results, but not in same sequence. It is not necessary to calculate all 3 roots of resolvent cubic. Any one non-zero root is sufficient to do the job. ==Two real and two complex roots== ===Example 1=== {{RoundBoxTop|theme=4}} [[File:0323_2curves02.png|thumb|400px|'''Graphs of quartic function with 2 real and 2 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 + 2x^3 + 18x^2 - 70x - 87</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,2,18,-70,-87 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> 264 -5568 -12208 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 8448 474112 -31002624 </syntaxhighlight> {{RoundBoxBottom}} Calculate one real root of cubic function: <math>y = g(x) = 64x^3 + 8448x^2 + 474112x -31002624.</math> <math>36</math> is one real root. Choose <math>U = 36.</math> Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 36 u1 = U**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) if V >= 0 : v = V**.5 else : v = 1j * (-V)**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =',x, '; sum =',sum ) </syntaxhighlight> <syntaxhighlight> x = 3.0 ; sum = 0.0 x = -1.0 ; sum = 0.0 x = (-2+5j) ; sum = 0j x = (-2-5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=4}} [[File:0321_2curves00.png|thumb|400px| Coefficient <math>d</math> of <math>g(x) = 0.</math> <math></math> ]] Calculate roots of: <math>y = f(x)</math> <math>= 3x^4 - 6x^3 - 41x^2 +44x - 189</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 3, -6, -41, 44, -189 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -2184, 0, -1229040 </syntaxhighlight> Notice that coefficient <math>B = 0.</math> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 1, -1092, 605376, 0 </syntaxhighlight> Notice that coefficient <math>S = 0.</math> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = x^3 - 1092x^2 + 605376x + 0.</math> Roots are <math>0, 546 \pm 554.3103823671355j.</math> Value <math>0</math> cannot be used because it will cause error <code>Divide by zero</code> at statement <code>V = -(A/2 + U) - B/(4*u)</code>. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 546+554.3103823671355j print ('\nU =',U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** 0.5 s1 = '\nu,v' print (s1,eval(s1)) for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> U = (546+554.3103823671355j) u,v ((25.729935131257832+10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (4.788322521876306 + 0j) ; sum = (1.9895196601282805e-13 + 0j) x = (0.5 + 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) u,v ((-25.729935131257832-10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (0.5 - 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) x = (-3.7883225218763052 + 0j) ; sum = (1.7053025658242404e-13 + 0j) </syntaxhighlight> Values of <math>x</math> are: <math>-3.7883225218763052, 4.788322521876306, 0.5 \pm 1.795283650280614j</math> <math></math><math></math> {{RoundBoxBottom}} ====Depressed quartic as quadratic==== In this example coefficient <math>B</math> of depressed quartic <math>= 0.</math> Therefore, resolvent cubic can be ignored and depressed quartic processed as quadratic in <math>T = t^2.</math> <math>t^4 - 2184t^2 + (0)t - 1229040</math> <math>T^2 - 2184T - 1229040</math> where <math>T = t^2.</math> Solutions of this quadratic are: <math>T_1, T_2 = 2648.1182474349434, -464.11824743494344</math> <syntaxhighlight lang=python> T1,T2 = 2648.1182474349434, -464.11824743494344 t1 = T1 ** 0.5; t2 = ((-T2) ** 0.5) * 1j for t in (t1,-t1,t2,-t2) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> x = 4.788322521876305 ; sum = -1.7053025658242404e-13 x = -3.788322521876305 ; sum = -1.7053025658242404e-13 x = (0.5 + 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) x = (0.5 - 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) </syntaxhighlight> or <syntaxhighlight> x = 0.5 ± 4.288322521876305, 0.5 ± 1.7952836502806138j </syntaxhighlight> With precision of 15, values of <math>x</math> are same as those shown above. When roots of quartic function are of form <code>p ± q, p ± r,</code> coefficient <math>B</math> of depressed function <math>= 0.</math> ==Four complex roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves03.png|thumb|400px|'''Graphs of quartic function with 4 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 - 20x^3 + 408x^2 + 2296x + 18020</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 4128 344064 9683200 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 64 132096 -86769664 -118380036096 </syntaxhighlight> {{RoundBoxBottom}} Calculate one root of cubic function: <math>y = g(x) = 64x^3 + 132096x^2 - 86769664x - 118380036096.</math> There are 3 real roots: <math>-2304, -784, 1024.</math> Choose <math>U = -784.</math> Negative <math>U</math> is chosen here to show that any 1 of the roots produces the correct result. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = -784 u1 = 1j * (-U)**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) v = V**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> # python expresses complex numbers with 'j'. x = (13+19j) ; sum = 0j x = (-3-5j) ; sum = 0j x = (13-19j) ; sum = 0j x = (-3+5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} =Quartic formula= The substitutions made above can be used to produce a formula for the solution of the quartic equation. See main articles [https://en.wikipedia.org/wiki/Quartic_equation#The_general_case "The general case" ] or [https://en.wikipedia.org/wiki/Quartic_function#General_formula_for_roots "General formula for roots."] {{RoundBoxTop|theme=2}} Both links above point to formula for equation <math>x^4 + ax^3 + bx^2 + cx + d = 0.</math> {{RoundBoxBottom}} Given quartic equation: <math>ax^4 + bx^3 + cx^2 + dx + e = 0,</math> calculate the 4 values of <math>x.</math> <math>x = \frac{-b + t}{4a}</math> where: {{RoundBoxTop|theme=2}} Coefficients of depressed quartic: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of resolvent cubic: <math>a_1 = P = 64</math> <math>b_1 = Q = 32A</math> <math>c_1 = R = 4A^2 - 16C</math> <math>d_1 = S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of depressed cubic: <math>A_1 = 9a_1 c_1 - 3b_1^2</math> <math>B_1 = 27a_1^2d_1 - 9a_1b_1c_1 + 2b_1^3</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of resolvent cubic: <math>C_1 = \frac{-A_1}{3} = b_1^2 - 3a_1 c_1</math> <math>\Delta = B_1^2 - 4C_1^3\ \dots\dots\ \Delta</math> may be negative. <math>\delta = \sqrt{\Delta}</math> <math>W = \frac{-B_1 + \delta}{2}</math> <math>w = \sqrt[3]{W}</math> <math>t_1 = w + \frac{C_1}{w}</math> <math>U = \frac{-b_1 + t_1}{3a_1}</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of quartic: <math>u = \sqrt{U}\ \dots\dots\ u</math> may be positive or negative. <math>V = -(\frac{A}{2} + U) - \frac{B}{4u}</math> <math>v = \sqrt{V}\ \dots\dots\ v</math> may be positive or negative. <math>t = u + v</math> {{RoundBoxBottom}} Formula above produces one value of <math>x.</math> Python code below utilizes <math>\pm \sqrt{U}</math> and <math>\pm \sqrt{V}</math> to produce 4 values of <math>t</math> and then, four values of <math>x.</math> ==An example:== {{RoundBoxTop|theme=4}} [[File:0330quartic01.png|thumb|400px|'''Graph of quartic function with 2 real roots.'''</br><math>Y</math> axis compressed for clarity.]] Calculate roots of <math>f(x) = 4x^4 + 4x^3 - 75x^2 - 776x - 1869.</math> <syntaxhighlight lang=python> # Python code. a,b,c,d,e = 4, 4, -75, -776, -1869 values_of_t = [ t # Coefficients of depressed quartic: for A in (16*a*c - 6*b*b,) for B in (64*a*a*d -32*a*b*c + 8*b*b*b,) for C in (256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b,) # Coefficients of resolvent cubic: for a1 in (64,) for b1 in (32*A,) for c1 in (4*A*A - 16*C,) for d1 in (-B*B,) for U in [ # The resolvent cubic: (-b1+t1)/(3*a1) # Coefficients of depressed resolvent cubic: for A1 in (9*a1*c1 - 3*b1*b1,) for B1 in (27*a1*a1*d1 - 9*a1*b1*c1 + 2*b1*b1*b1,) # One root of resolvent cubic: for C1 in (-A1/3,) for Δ in (B1*B1 - 4*C1*C1*C1,) for δ in (Δ**0.5,) for W in ((-B1 + δ)/2,) for w in (W**(1/3),) for t1 in (w + C1/w,) # See note below. ] # Prepare to calculate 4 values of t. for u1 in (U**.5,) for v1 in ( -(A/2 + U) ,) # Calculate 4 values of t. for u in (u1, -u1,) for V in ( v1 - B/(4*u),) for v in (V**.5,) for t in (u+v, u-v) ] print ('values_of_t =', values_of_t) </syntaxhighlight> {{RoundBoxTop|theme=4}} <syntaxhighlight> values_of_t = [116, -44, (-36+64j), (-36-64j)] </syntaxhighlight> Because <math>f(t)</math> is a depressed quartic function, sum of four <code>values_of_t</code> <math>= 116 - 44 - 36(2) = 0.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> # Python code. # Calculate 4 separate roots. values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] print ('values_of_x =', values_of_x) </syntaxhighlight> <syntaxhighlight lang=python> values_of_x = [7, -3, (-2.5 + 4j), (-2.5 - 4j)] </syntaxhighlight> In python the imaginary part of a complex number is shown with <math>j</math> instead of <math>i.</math> If <math>A == B == C == 0,\ f(x)</math> contains 4 equal roots and <math>x = \frac{-b}{4a}.</math> If <math>f(x)</math> contains 3 or more equal roots, statement <code>for t1 in (w + C1/w,)</code> fails with divisor <math>w = 0.</math> Before using this formula, check for equal roots as in [[Quartic_function#Exactly_3_equal_roots | "Exactly 3 equal roots"]] above. {{RoundBoxTop|theme=1}} Values displayed above have been edited slightly. Actual calculated values were: <syntaxhighlight> values_of_x = [7.000000000000001, -3.0000000000000044, (-2.499999999999998+4.000000000000001j), (-2.4999999999999987-4.000000000000001j)] </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ==In practice== {{RoundBoxTop|theme=5}} The following Python code implements the quartic formula. However, under statement <code>if B4 == 0 :</code> there is code that processes the depressed quartic as a quadratic in <math>T = t^2.</math> This ensures that execution of formula does not fail with error <code>Divide by zero</code> at statement <code>for V in ( v1 - B4/u,).</code> {{RoundBoxTop|theme=4}} <syntaxhighlight lang=python> # python code import cmath cxSqrt = cmath.sqrt # Square root of complex number. def rootsOfQuartic (abcde) : ''' x1,x2,x3,x4 = rootsOfQuartic ((a,b,c,d,e)) Each member of input must be int or float or Decimal object. Int or Decimal object in input is quietly converted to float. Output may be None. ''' def formatResults (x1x2x3x4) : ''' This function improves appearance of results. (8 + 0j) becomes 8.0 ''' values_of_x = list (x1x2x3x4) for p in (0,1,2,3) : v = values_of_x[p] if isinstance (v, complex) and (v.imag == 0) : values_of_x[p] = v.real return values_of_x status = 0 try : a,b,c,d,e = [ float(v) for v in abcde ] except : status = 1 if status : print ('rootsOfQuartic () 1: Error creating coefficients a,b,c,d,e.') return None if a == 0 : print ('rootsOfQuartic () 2: Coefficient a must be non-zero.') return None # Coefficients of depressed quartic, modified. # A = 16*a*c - 6*b*b A2 = 8*a*c - 3*b*b # B = 64*a*a*d - 32*a*b*c + 8*b*b*b B4 = 16*a*a*d - 8*a*b*c + 2*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b if B4 == 0 : # B = 0. # Result returned from this section is type tuple, indicating that coefficient B4 = 0. if A2==C==0 : # 4 equal roots. root = -b/(4*a) return tuple(formatResults((root,root,root,root))) # t**4 + At**2 + (0)t + C # Depressed quartic is quadratic in T: # T**2 + AT + C where T = t**2 # T**2 + 2(A2)T + C where A = 2(A2) # -2(A2) +/- (4(A2)(A2) - 4C)**0.5 # T = -------------------------------- = -A2 +/- ((A2)(A2) - C)**0.5 # 2 disc = A2*A2 - C if disc >= 0 : root = disc ** 0.5 else : root = ((-disc) ** 0.5) * 1j T1 = -A2 - root ; T2 = -A2 + root t1 = cxSqrt(T1) ; t2 = cxSqrt(T2) values_of_t = (t1,-t1,t2,-t2) values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] return tuple(formatResults(values_of_x)) # B4 is non-zero. Therefore, all of (S, U, u) are non-zero. P,Q,R,S = 1, A2, (A2*A2 - C)/4, -B4*B4/4 # str1 = 'P,Q,R,S' ; print (str1, eval(str1)) U = oneRootOfCubic((P,Q,R,S)) # Resolvent cubic. if U > 0 : sqrtU = U ** 0.5 elif U == 0 : # This should not happen. print ('rootsOfQuartic () 3: Internal error.') return None else : sqrtU = ((-U) ** 0.5) * 1j v1 = -(A2+U) values_of_t = [ t for u in (sqrtU, -sqrtU) for V in ( v1 - B4/u, ) for v in ( cxSqrt(V), ) for t in (u+v, u-v) ] values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] # Result returned from this section is type list, indicating that coefficient B4 != 0. return formatResults(values_of_x) </syntaxhighlight> For function <code>oneRootOfCubic()</code> see [[Cubic_function#In_practice | Cubic_function: In_practice.]] {{RoundBoxBottom}} ===Examples=== {{RoundBoxTop|theme=2}} Python function <code>equalRoots()</code> below implements <code>status</code> as presented under [https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above. <syntaxhighlight lang=python> # python code def equalRoots(abcde) : ''' This function returns True if quartic function contains at least 2 equal roots. ''' a,b,c,d,e = abcde aa = a*a ; aaa = aa*a bb = b*b ; bbb = bb*b ; bbbb = bb*bb cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd ee = e*e ; eee = ee*e ; eeee = ee*ee v1 = ( +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd +20*bb*ccc*dd*e ) v2 = ( -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd ) return (v1+v2) == 0 t1 = ( ((1, -1, -19, -11, 30), '4 unique, real roots.'), ((4, 4,-119, -60, 675), '4 unique, real roots, B4 = 0.'), ((1, 6, -48,-182, 735), '2 equal roots.'), ((1,-12, 50, -84, 45), '2 equal roots. B4 = 0.'), ((1,-20, 146,-476, 637), '2 equal roots, 2 complex roots.'), ((1,-12, 58,-132, 117), '2 equal roots, 2 complex roots. B4 = 0.'), ((1, -2, -36, 162, -189), '3 equal roots.'), ((1,-20, 150,-500, 625), '4 equal roots. B4 = 0.'), ((1, -6, -11, 60, 100), '2 pairs of equal roots, B4 = 0.'), ((4, 4, -75,-776,-1869), '2 complex roots.'), ((1,-12, 33, 18, -208), '2 complex roots, B4 = 0.'), ((1,-20, 408,2296,18020), '4 complex roots.'), ((1,-12, 83, -282, 442), '4 complex roots, B4 = 0.'), ((1,-12, 62,-156, 169), '2 pairs of equal complex roots, B4 = 0.'), ) for (abcde, comment) in t1 : print () fourRoots = rootsOfQuartic (abcde) print (comment) print (' Coefficients =', abcde) print (' Four roots =', fourRoots) print (' Equal roots detected:', equalRoots(abcde)) # Check results. a,b,c,d,e = abcde for x in fourRoots : # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0 sum = (a*x**4 + b*x**3 + c*x**2 + d*x + e) if sum : # Create exception. 1/0 </syntaxhighlight> <syntaxhighlight> 4 unique, real roots. Coefficients = (1, -1, -19, -11, 30) Four roots = [5.0, 1.0, -2.0, -3.0] Equal roots detected: False 4 unique, real roots, B4 = 0. Coefficients = (4, 4, -119, -60, 675) Four roots = (2.5, -3.0, 4.5, -5.0) Equal roots detected: False 2 equal roots. Coefficients = (1, 6, -48, -182, 735) Four roots = [5.0, 3.0, -7.0, -7.0] Equal roots detected: True 2 equal roots. B4 = 0. Coefficients = (1, -12, 50, -84, 45) Four roots = (3.0, 3.0, 5.0, 1.0) Equal roots detected: True 2 equal roots, 2 complex roots. Coefficients = (1, -20, 146, -476, 637) Four roots = [7.0, 7.0, (3+2j), (3-2j)] Equal roots detected: True 2 equal roots, 2 complex roots. B4 = 0. Coefficients = (1, -12, 58, -132, 117) Four roots = ((3+2j), (3-2j), 3.0, 3.0) Equal roots detected: True 3 equal roots. Coefficients = (1, -2, -36, 162, -189) Four roots = [3.0, 3.0, 3.0, -7.0] Equal roots detected: True 4 equal roots. B4 = 0. Coefficients = (1, -20, 150, -500, 625) Four roots = (5.0, 5.0, 5.0, 5.0) Equal roots detected: True 2 pairs of equal roots, B4 = 0. Coefficients = (1, -6, -11, 60, 100) Four roots = (5.0, -2.0, 5.0, -2.0) Equal roots detected: True 2 complex roots. Coefficients = (4, 4, -75, -776, -1869) Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)] Equal roots detected: False 2 complex roots, B4 = 0. Coefficients = (1, -12, 33, 18, -208) Four roots = ((3+2j), (3-2j), 8.0, -2.0) Equal roots detected: False 4 complex roots. Coefficients = (1, -20, 408, 2296, 18020) Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)] Equal roots detected: False 4 complex roots, B4 = 0. Coefficients = (1, -12, 83, -282, 442) Four roots = ((3+5j), (3-5j), (3+2j), (3-2j)) Equal roots detected: False 2 pairs of equal complex roots, B4 = 0. Coefficients = (1, -12, 62, -156, 169) Four roots = ((3+2j), (3-2j), (3+2j), (3-2j)) Equal roots detected: True </syntaxhighlight> When description contains note <math>B4 = 0,</math> depressed quartic was processed as quadratic in <math>t^2.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Two Conic Sections= Examples of conic sections include: ellipse, circle, parabola and hyperbola. This section presents examples of two conic sections, circle and ellipse, and how to calculate the coordinates of the point/s of intersection, if any, of the two sections. Let one section with name <math>ABCDEF</math> have equation <math>Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.</math> Let other section with name <math>abcdef</math> have equation <math>ax^2 + by^2 + cxy + dx + ey + f = 0.</math> Because there can be as many as 4 points of intersection, a special "resolvent" quartic function is used to calculate the <math>x</math> coordinates of the point/s of intersection. Coefficients of associated "resolvent" quartic are calculated as follows: <syntaxhighlight lang=python> # python code def intersection_of_2_conic_sections (abcdef, ABCDEF) : ''' A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF) where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function: y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_ ''' A,B,C,D,E,F = ABCDEF a,b,c,d,e,f = abcdef G = ((-1)*(B)*(a) + (1)*(A)*(b)) H = ((-1)*(B)*(d) + (1)*(D)*(b)) I = ((-1)*(B)*(f) + (1)*(F)*(b)) J = ((-1)*(C)*(a) + (1)*(A)*(c)) K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c)) L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c)) M = ((-1)*(E)*(f) + (1)*(F)*(e)) g = ((-1)*(C)*(b) + (1)*(B)*(c)) h = ((-1)*(E)*(b) + (1)*(B)*(e)) A_ = (J)*(g) + (G)*(G) B_ = (J)*(h) + (K)*(g) + (2)*(G)*(H) C_ = (K)*(h) + (L)*(g) + (2)*(G)*(I) + (H)*(H) D_ = (L)*(h) + (M)*(g) + (2)*(H)*(I) E_ = (M)*(h) + (I)*(I) str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_) print (str1) return A_,B_,C_,D_,E_ </syntaxhighlight> == With no common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections06.png|thumb|400px|'''Two conic sections with no common point.''' </br> Resolvent quartic function (black curve) has no real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 80 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.98x^2 + 6850.532x - 22836.7009.</math> <math>f(x)</math> has no real roots. Therefore, there is no point of intersection. {{RoundBoxBottom}} ==With one common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections05.png|thumb|400px|'''Two conic sections with one common point.''' </br> Resolvent quartic function (black curve) has two equal, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 73 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.84x^2 + 7456.48x - 24355.36.</math> Roots of <math>f(x)</math> are: <math>(-14-22.978250586152114j),</math> <math>(-14+22.978250586152114j),</math> <math>5.8, 5.8.</math> <math>f(x)</math> has 2 equal, real roots at <math>x = 5.8,</math> effectively 1 real root where <math>x = 5.8</math> Therefore, there is one point of intersection where <math>x = 5.8.</math> {{RoundBoxBottom}} ==With two common points== ===Example 1=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections04.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two unique, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 64 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.66x^2 + 8235.556x - 26681.1841.</math> Roots of <math>f(x)</math> are: <math>(-14.361578825892241-23.341853011785357j),</math> <math>(-14.361578825892241+23.341853011785357j),</math> <math>4.59885619413921, 7.72430145764527.</math> <math>f(x)</math> has 2 unique, real roots at <math>x = 4.59885619413921, 7.72430145764527.</math> Therefore, there are two points of intersections where <math>x = 4.59885619413921, 7.72430145764527.</math> {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections01.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two pairs of equal roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 18.8x - 1.6y + 53 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 + 37.6x^3 - 504.24x^2 + 2835.04x - 5685.16.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 13, 13.</math> <math>f(x)</math> has 2 pairs of equal roots at <math>x = 5.8, 13,</math> effectively 2 real roots. Therefore, there are two points of intersection where <math>x = 5.8, 13.</math> {{RoundBoxBottom}} ==With 3 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections03.png|thumb|400px|'''Two conic sections with three common points.''' </br> Resolvent quartic function (black curve) has one pair of equal roots and 2 unique, real roots. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 17.6x - 3.2y + 55 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 6.83589838486224, 13.7641016151377 .</math> <math>f(x)</math> has 1 pair of equal roots at <math>x = 5.8</math> and 2 unique, real roots at <math>x = 6.83589838486224, 13.7641016151377,</math> effectively 3 real roots. Therefore, there are three points of intersection where <math>x = 5.8, 6.83589838486224, 13.7641016151377.</math> {{RoundBoxBottom}} ==With 4 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections02.png|thumb|400px|'''Two conic sections with four common points.''' </br> Resolvent quartic function (black curve) has 4 unique, real roots. </br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 18.8x - 1.6y + 62.99 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x)</math><math>= -x^4 + 37.6x^3 - 504.4398x^2 + 2838.79624x - 5544.61147921.</math> Roots of <math>f(x)</math> are:<math>4.36661032156283,</math><math>8.77936456353008,</math> <math>10.0206354364699,</math><math>14.4333896784371.</math> <math>f(x)</math> has 4 real roots as shown above. Therefore, there are four points of intersection where <math>x = 4.36661032156283,</math> <math>8.77936456353008,</math><math>10.0206354364699,</math><math>14.4333896784371.</math> {{RoundBoxBottom}} ===Values of y=== Section above calculates <math>x</math> coordinates of the four common points. This section calculates <math>y</math> coordinates. Here is the lazy way to do it: {{RoundBoxTop|theme=2}} [[File:0818ResQuartic.png|thumb|400px|'''Graphs of 2 curves, red and blue, with coefficients of <math>x,y</math> reversed.''' </br> Roots of "resolvent" quartic, black curve, are interpreted as values of <math>y.</math> ]] <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 # Coefficients of x,y are reversed. abcde = intersection_of_2_conic_sections ((b,a,0,e,d,f), (B,A,C,E,D,F)) values_of_y = rootsOfQuartic (abcde) print (values_of_y) </syntaxhighlight> <syntaxhighlight> [1.6215767432213488, -0.021576743221348723, 5.862095579401657, -4.262095579401658] </syntaxhighlight> By inspection of the diagram above, it is obvious that: <syntaxhighlight lang=python> # python code four_common_points = ( (4.36661032156283, 1.6215767432213488), (8.77936456353009, -4.262095579401658), (10.0206354364699, 5.862095579401657), (14.4333896784372, -0.021576743221348723), ) </syntaxhighlight> Check results: <syntaxhighlight lang=python> # python code def sum_zero (input) : ''' This function calculates sum of all values in input. If sum is very close to 0, this function returns 0. ''' sum = 0 for v in input : sum += v if sum == 0 : return sum tolerance = 1e-14 max = sorted([ abs(v) for v in input ])[-1] if abs(sum) < tolerance*max : return 0 return sum # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 for (x,y) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum1 = sum_zero(values) if sum1 : print ('Error: sum1 =',sum1) values = a*x**2, b*y**2, d*x, e*y, f sum2 = sum_zero(values) if sum2 : print ('Error: sum2 =',sum2) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. Check results with coefficients of <math>x, y</math> reversed: <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = B,A,C,E,D,F # The circle: a,b,c,d,e,f = b,a,c,e,d,f for (y,x) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum3 = sum_zero(values) if sum3 : print ('Error: sum3 =',sum3) values = a*x**2, b*y**2, d*x, e*y, f sum4 = sum_zero(values) if sum4 : print ('Error: sum4 =',sum4) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. {{RoundBoxBottom}} =Links to related topics= {{RoundBoxTop|theme=1}} [[Cubic_function#Cubic_formula | "Cubic formula"]] [[Complex_square_root | "Complex square root" ]] {{RoundBoxBottom}} [[Category:Mathematics]] r4hszs9jwct2ev7zzh1c79lver4rxaq 2844183 2844182 2026-11-03T11:34:26Z ThaniosAkro 2805358 /* With 4 common points */ 2844183 wikitext text/x-wiki The quartic function is the bridge between the cubic function and more advanced functions such as the quintic and sextic. =Objective= {{RoundBoxTop|theme=3}} [[file:Books-aj.svg_aj_ashton_01f.png|right|100px]] * Present quartic function and quartic equation. * Introduce the concept of roots of equal absolute value. * Show how to predict and calculate equal roots, techniques that will be useful when applied to higher order functions. * Simplify the depressed quartic. * Show that the quartic equation is effectively solved when at least one root is known. * Present the "resolvent" cubic function. * Show how to derive and use the quartic formula. {{RoundBoxBottom}} =Lesson= {{RoundBoxTop|theme=2}} ==Introduction== The quartic function is the sum of powers of <math>x</math> from <math>0</math> through <math>4</math>: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx^1 + ex^0</math> usually written as: <math>y = f(x) = ax^4 + bx^3 + cx^2 + dx + e.</math> If <math>e == 0</math> the function becomes <math>x(ax^3 + bx^2 + cx + d).</math> Within this page we'll say that: * both coefficients <math>a, e</math> must be non-zero, * coefficient <math>a</math> must be positive (simply for our convenience), * all coefficients must be real numbers, accepting that the function may contain complex roots. The quartic equation is the quartic function equated to zero: <math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>. Roots of the function are values of <math>x</math> that satisfy the quartic equation. {{RoundBoxTop|theme=1}} Because the function is "quartic" (maximum power of <math>x</math> is <math>4</math>), the function contains exactly <math>4</math> roots, an even number of complex roots and an even number of real roots. Other combinations of real and complex roots are possible, but they produce complex coefficients. {{RoundBoxBottom}} {{RoundBoxTop|theme=7}} [[File:0312quartic01a.png|thumb|400px|'''Graph of typical quartic function showing minima and maximum.''']] The figure shows a typical quartic function. The function crosses the <math>X</math> axis in 4 different places. The function has 4 roots: <math>(-2,0), (1,0), (5,0), (10,0).</math> This function contains one local minimum, one local maximum and one absolute minimum. There is no absolute maximum. Because the function contains one absolute minimum: * If abs(<math>x</math>) is very large, <math>f(x)</math> is always positive. * If absolute minimum is above <math>X</math> axis, curve does not cross <math>X</math> axis and function contains only complex roots. * There is always at least one point where the curve is parallel to <math>X</math> axis. The curve is never parallel to the <math>Y</math> axis. For any real value of <math>x</math> there is always a real value of <math>y.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0312quartic02.png|thumb|400px|'''When coefficient d is missing, there is a stationary point at x = 0.''']] If coefficient <math>d</math> is missing, the quartic function becomes <math>y = ax^4 + bx^3 + cx^2 + e,</math> and <math>y' = 4ax^3 + 3bx^2 + 2cx = x(4ax^2 + 3bx + 2c).</math> For a stationary point <math>y' = x(4ax^2 + 3bx + 2c) = 0.</math> When coefficient <math>d</math> is missing, there is always a stationary point at <math>x = 0.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} [[File:0319quartic02.png|thumb|400px|'''Graph of quartic function that is quadratic in <math>x^2</math>.'''</br>Because coefficient <math>d</math> is missing, there is a stationary point where <math>x = 0.</math>]] If coefficients <math>b, d</math> are missing, the quartic function becomes a quadratic in <math>x^2.</math> The curve (red line) in diagram has equation: <math>y = f(x) = \frac{x^4 - 13x^2 + 36}{5}</math> The quartic equation may be solved as: <math>X^2 - 13X + 36 = 0</math> where <math>X = x^2</math> or <math>x = \sqrt{X}.</math> <math>X = 4</math> or <math>X = 9.</math> <math>x = \pm 2</math> or <math>x = \pm 3.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0312quartic03.png|thumb|400px| ]] The quartic function may be expressed as <math>x = ay^4 + by^3 + cy^2 + dy + e.</math> Unless otherwise noted, references to "quartic function" on this page refer to function of form <math>y = ax^4 + bx^3 + cx^2 + dx + e.</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} [[File:0318quartic01.png|thumb|400px|'''Graph of quartic function with coefficient a negative.'''</br>There is no absolute minimum.]] Coefficient <math>a</math> may be negative as shown in diagram. As <code>abs</code><math>(x)</math> increases, the value of <math>f(x)</math> is dominated by the term <math>-ax^4.</math> When <code>abs</code><math>(x)</math> is very large, <math>f(x)</math> is always negative. Unless stated otherwise, any reference to "quartic function" on this page will assume coefficient <math>a</math> positive. {{RoundBoxBottom}} {{RoundBoxTop|theme=1}} [[File:0502quartic01.png|thumb|400px|'''Graph of quartic function with coefficient b missing.''' </br> Sum of roots is <math>0.</math> </br> <math>Y</math>axis compressed for clarity. ]] When sum of roots is <math>0,</math> coefficient <math>b = 0.</math> In the diagram, roots of <math>f(x)</math> are <math>-5, -4, 2, 7.</math> Sum of roots <math>= 0.</math> Therefore coefficient <math>b = 0.</math> {{RoundBoxBottom}} ==Function as product of linear function and cubic== {{RoundBoxTop|theme=3}} [[File:0313quartic01.png|thumb|400px|'''Graphs of quartic function and associated cubic function.''']] When <math>p</math> is a root of the function, the function may be expressed as: <math>(x - p)( Ax^3 + Bx^2 + Cx + D )</math> where <math>A = a;\ B = Ap + b;\ C = Bp + c;\ D = Cp + d.</math> When one real root <math>p</math> is known, the other three roots may be calculated as roots of the cubic function <math>Ax^3 + Bx^2 + Cx + D.</math> In the diagram the quartic function has equation: <math>y = \frac{ x^4 - 23x^3 + 163x^2 - 393x + 252 }{ 48 }.</math> It is known that <math>3</math> is a root of this function. The associated cubic has equation: <math>y = \frac { x^3 - 20x^2 + 103x - 84} { 48 }</math> The 2 curves coincide at points <math>(1, 0),\ (7, 0),\ (12, 0),</math> the three points that are roots of both functions. <math></math> {{RoundBoxBottom}} ==Function defined by 5 points== {{RoundBoxTop|theme=2}} [[File:0313quartic02.png|thumb|400px|'''Figure 1. Quartic function defined by 5 points''' </br>Any 5 points on the curve may be used to define the function.]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 5 unique points on the curve. For example, let us choose the five points: <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code points = (-5,0), (-2,0), (1,0), (3,-6), (6,2) L11 = [] for point in points : x,y = point L11 += [[x*x*x*x, x*x*x, x*x, x, 1, -y]] print (L11) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ 625.0, -125.0, 25.0, -5.0, 1.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 0.0], # [ 1.0, 1.0, 1.0, 1.0, 1.0, 0.0], # matrix supplied to function solveMbyN() below. [ 81.0, 27.0, 9.0, 3.0, 1.0, 6.0], # 5 rows by 6 columns. [1296.0, 216.0, 36.0, 6.0, 1.0, -2.0]] # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L11) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.02651515151515152, 0.004545454545454542, -0.847727272727273, -0.728787878787879, 1.5454545454545459) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by the 5 points <math>(-5,0), (-2,0), (1,0), (3,-6), (6,2)</math> is <math>y = \frac{0.875 x^4 + 0.15 x^3 - 27.975 x^2 - 24.05 x + 51}{33}.</math> {{RoundBoxBottom}} ==Function defined by 3 points and 2 slopes== {{RoundBoxTop|theme=2}} [[File:0314quartic03.png|thumb|400px|'''Figure 1. Quartic function defined by 3 points and 2 slopes.''' </br>Any 3 points on the curve and the slopes at any 2 of these points may be used to define the function.</br></br>Slope at <math>(-2, -2)\ =</math> slope at <math>(6, -4)\ =\ 0.</math>]] Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function. See Figure 1. The quartic function may be defined by any 3 unique points on the curve and the slopes at any 2 of these points. For example, let us choose the three points: <math>(-2, -2), (6, -4), (4, 1)</math> At point <math>(-2, -2)</math> slope is <math>0.</math> At point <math>(6, -4)</math> slope is <math>0.</math> Rearrange the standard quartic function to prepare for the calculation of <math>a,b,c,d,e:</math> <math>x^4 a + x^3 b + x^2 c + x d + 1e - y = 0.</math> Rearrange the standard cubic function of slope to prepare for the calculation of <math>a,b,c,d,e:</math> <math> 4 x^3 a + 3 x^2 b + 2 x c + 1 d + 0 e - s = 0.</math> For function <code>solveMbyN</code> see [[Solving_simultaneous_equations#Solving_M_by_(M+1) | "Solving simultaneous equations" ]]. {{RoundBoxTop|theme=3}} <syntaxhighlight lang=python> # python code def makeEntry(input) : x,y,s = ( tuple(input) + (None,) )[:3] L1 = [] if s != None : L2 = [ float(v) for v in [4*x*x*x, 3*x*x, 2*x, 1, 0, -s] ] L1 += [ L2 ] L2 = [ float(v) for v in [x*x*x*x, x*x*x, x*x, x, 1, -y]] L1 += [ L2 ] return L1 t1 = ( (-2,-2, 0), # point (-2, -2) with slope 0. (6,-4, 0), # point (6, -4) with slope 0. (4,1), # point (4, 1) ) L1 = [] for v in t1 : L1 += makeEntry ( v ) print (L1) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> [[ -32.0, 12.0, -4.0, 1.0, 0.0, 0.0], # [ 16.0, -8.0, 4.0, -2.0, 1.0, 2.0], # [ 864.0, 108.0, 12.0, 1.0, 0.0, 0.0], # matrix supplied to function solveMbyN() below. [1296.0, 216.0, 36.0, 6.0, 1.0, 4.0], # 5 rows by 6 columns. [ 256.0, 64.0, 16.0, 4.0, 1.0, -1.0]]. # </syntaxhighlight> {{RoundBoxBottom}} <syntaxhighlight lang=python> # python code output = solveMbyN(L1) print (output) </syntaxhighlight> {{RoundBoxTop|theme=3}} <syntaxhighlight> # 5 coefficients a, b, c, d, e: (0.03255208333333339, -0.2526041666666665, -0.3072916666666667, 2.84375, 2.375) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} Quartic function defined by three points and two slopes is: <math>y = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> ===Associated cubic functions=== {{RoundBoxTop|theme=3}} ====When p == -2==== {{RoundBoxTop|theme=1}} [[File:0314_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = -2.''' </br>In this case roots of associated cubic include x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == -2,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 15.25 x^2 + 15.75 x + 105 } {48}.</math> Three blue vertical lines show 3 values of <math>x</math> where <math>g(x) = 0</math> and <math>f(x) = f(-2)</math> In this case roots of <math>g(x)</math> include <math>x = p.</math> {{RoundBoxBottom}} ====When p == 5==== {{RoundBoxTop|theme=1}} [[File:0314_2curves02.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 5.''' </br>In this case the one root of associated cubic excludes x = p.]] Quartic function is: <math>y = f(x) = \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 5,</math> associated cubic function is : <math>y = g(x) = \frac{ 1.5625 x^3 - 4.3125 x^2 - 36.3125 x - 45.0625 } {48}.</math> Two blue vertical lines show 2 values of <math>x</math> where <math>f(x) = f(5)</math> In this case the one root of <math>g(x)</math> excludes <math>x = p.</math> {{RoundBoxBottom}} ====When p == 6==== {{RoundBoxTop|theme=1}} [[File:0315_2curves01.png|thumb|400px|'''Figure 1. Quartic function and associated cubic function when</br>p = 6.''' </br>In this case the one root of associated cubic includes x = p.]] Quartic function is: <math>y = f(x) </math><math>= \frac{ 1.5625 x^4 -12.125 x^3 -14.75 x^2 + 136.5 x + 114.0 } {48}.</math> When <math>p == 6,</math> associated cubic function is: <math>y = g(x) </math><math>= \frac{ 1.5625 x^3 - 2.75 x^2 - 31.25 x - 51 } {48}.</math> One blue vertical line shows 1 value of <math>x</math> where <math>f(x) = f(6)</math> In this case the one root of <math>g(x)</math> includes <math>x = p.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} ==Examples== {{RoundBoxTop|theme=2}} ===Quartic with 2 stationary points=== {{RoundBoxTop|theme=1}} [[File:0315_2statPoints02.png|thumb|400px|'''Graph of quartic function with 2 stationary points.''']] In the diagram the red line represents quartic function <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 2 roots of <math>g(x),\ -1</math> and <math>\frac{-1}{4}</math> show that <math>f(x)</math> has stationary points at <math>x = -1</math> and <math>x = -0.25.</math> {{RoundBoxBottom}} ===Quartic with 1 stationary point=== {{RoundBoxTop|theme=1}} [[File:0315_1statPoint.png|thumb|400px|'''Graph of quartic function with 1 stationary point.''']] In the diagram the red line represents quartic function <math>y = f(x) = \frac{x^4 + 3x^3 + 3x^2 + 3x}{2}</math> The grey line <math>g(x)</math> is the first derivative of <math>f(x).</math> The 1 root of <math>g(x),\ -1.607</math> (approx.), shows that <math>f(x)</math> has 1 stationary point where <math>g(x) = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =First and second derivatives= {{RoundBoxTop|theme=1}} ==Points of inflection== {{RoundBoxTop|theme=2}} [[File:0317_3curves01.png|thumb|400px|'''Graphs of quartic function and first two derivatives.'''</br>Dotted portion of black line shows where f(x) is always concave down.</br>Dotted portion of red line shows where g(x) is decreasing.</br>Dotted portion of blue line shows where h(x) is negative.</br>When h(x) is negative, f(x) is concave down.</br>When h(x) is positive, f(x) is concave up.]] In the diagram the black line has equation: <math>y = f(x) = \frac{x^4 - 16x^3 + 42x^2 + 12x + 4}{144}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{x^3 - 12x^2 + 21x + 3}{36}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{x^2 - 8x + 7}{12}.</math> When <math>x < x_1:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. When <math>x == x_1:</math> * <math>y'</math> is at a local maximum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between up and down. When <math>x_1 < x < x_2:</math> * <math>y'</math> is decreasing. * <math>y''</math> is negative. * <math>f(x)</math> is always concave down. When <math>x == x_2:</math> * <math>y'</math> is at a local minimum. * <math>y'' = 0.</math> * Concavity of <math>f(x)</math> is between down and up. When <math>x_2 < x:</math> * <math>y'</math> is increasing. * <math>y''</math> is positive. * <math>f(x)</math> is always concave up. {{RoundBoxTop|theme=1}} The roots of <math>h(x): x_1 = 1, x_2 = 7.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At point <math>p_1</math> concavity of <math>f(x)</math> changes from up to down. At point <math>p_2</math> concavity of <math>f(x)</math> changes from down to up. The points <math>p_1, p_2</math> (the <math>X</math> coordinates of which are roots of <math>h(x)</math>) are the '''points of inflection''' of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Maxima and minima== {{RoundBoxTop|theme=2}} [[File:0317_3curves03.png|thumb|400px|'''Graphs of quartic function and first two derivatives showing maximum and minima.'''</br> Point <math>p_{2}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{2}</math> is concave down. Point <math>p_{2}</math> is local maximum. </br> Point <math>p_{1}</math> on <math>f(x)</math> is a stationary point. <math>f(x)</math> at point <math>p_{1}</math> is concave up. Point <math>p_{1}</math> is local minimum. </br> Similarly, point <math>p_{3}</math> is local minimum. ]] In the diagram the black line has equation: <math>y = f(x) = \frac{1.5625x^4 - 12.125x^3 - 14.75x^2 + 136.5x + 114}{48}.</math> The first derivative, the red line, has equation: <math>y' = g(x) = \frac{6.25x^3 - 36.375x^2 - 29.5x + 136.5}{48}.</math> The second derivative, the blue line, has equation: <math>y'' = h(x) = \frac{18.75x^2 - 72.75x - 29.5}{48}.</math> Roots of <math>g(x):\ x_1 = -2;\ x_2 = 1.82;\ x_3 = 6.</math> Let point <math>p_1</math> on <math>f(x)</math> have coordinates <math>(x_1, f(x_1)).</math> At <math>x_1\ h(x_1)</math> is positive. Point <math>p_1</math> is a stationary point and <math>f(x)</math> at <math>p_1</math> is concave up. Point <math>p_1</math> is a local minimum. Let point <math>p_2</math> on <math>f(x)</math> have coordinates <math>(x_2, f(x_2)).</math> At <math>x_2\ h(x_2)</math> is negative. Point <math>p_2</math> is a stationary point and <math>f(x)</math> at <math>p_2</math> is concave down. Point <math>p_2</math> is a local maximum. Let point <math>p_3</math> on <math>f(x)</math> have coordinates <math>(x_3, f(x_3)).</math> At <math>x_3\ h(x_3)</math> is positive. Point <math>p_3</math> is a stationary point and <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is a local minimum. {{RoundBoxBottom}} ==Quartic with 2 stationary points== {{RoundBoxTop|theme=3}} [[File:0318_3curves01.png|thumb|400px|'''Graph of quartic function with 2 stationary points and first 2 derivatives.''' </br> Black line: <math>y = f(x) = 4(x^4 + 3x^3 + 3x^2 + x) - 1.</math> </br> Red line: <math>y' = g(x) = 4(4x^3 + 9x^2 + 6x + 1).</math> </br> <math>y'' = h(x) = 4(12x^2 + 18x + 6)</math> </br> Blue line: <math>y'' = 4(2x^2 + 3x + 1) = \frac{h(x)}{6}.</math> </br> Dotted portion of black line shows where <math>f(x)</math> is concave down. ]] In the diagram, point <math>p_1</math> on <math>f(x)</math> has coordinates <math>(x_1, f(x_1)).</math> Similarly, points <math>p_2, p_3</math> have coordinates <math>(x_2, f(x_2)),\ (x_3, f(x_3)).</math> <math>y'</math> has roots: <math>x_1 = -1;\ x_3 = -0.25.</math> Points <math>p_1, p_3</math> are stationary points. <math>y''</math> has roots: <math>x_1 = -1;\ x_2 = -0.5.</math> Points <math>p_1, p_2</math> are points of inflection. At point <math>p_3\ y''</math> is positive. <math>f(x)</math> at <math>p_3</math> is concave up. Point <math>p_3</math> is local minimum. {{RoundBoxTop|theme=2}} Summary: * Point <math>p_1</math> is both stationary point and point of inflection. * Point <math>p_2</math> is point of inflection. * Point <math>p_3</math> is both stationary point and local minimum. {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =The simplest quartic function= {{RoundBoxTop|theme=2}} [[File:0320quartic01.png|thumb|400px|'''Graph of simplest quartic function.''' </br> Point <math>p_1\ (-1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_2\ (1.1, 0)</math> is a root of <math>f(x).</math> </br> Point <math>p_0\ (0, -1.1^4)</math> is <math>Y</math> intercept of <math>f(x).</math> ]] The simplest quartic function has coefficients <math>b = c = d = 0.</math> Red line in diagram has equation: <math>y = f(x) = x^4 - 1.1^4</math> First derivative (not shown) of <math>f(x):\ y' = g(x) = 4x^3.</math> When <math>x == 0,\ g(x) = 0.</math> There is a stationary point on <math>f(x)</math> when <math>x == 0,</math> point <math>p_0.</math> Second derivative (not shown) of <math>f(x):\ y'' = h(x) = 12x^2.</math> When <math>x == 0,\ h(x) = 0.</math> There is a point of inflection on <math>f(x)</math> when <math>x == 0.</math> For every non-zero value of <math>x,\ h(x)</math> is positive. To left and right of point <math>p_0,\ f(x)</math> is always concave up. Point <math>p_0</math> is both local minimum and absolute minimum. * Point <math>p_0</math> is stationary point and point of inflection and absolute minimum. Curve <math>f(x)</math> is useful for finding the fourth root of a real number. Solve: <math>x = N^{\frac{1}{4}}.</math> <math>x^4 = N.</math> <math>x^4 - N = 0.</math> This is equivalent to finding a root of function <math>y = j(x) = x^4 - N.</math> If you use Newton's method to find a root of <math>j(x),</math> this would be more efficient than solving <math>x = \sqrt{\sqrt{N}}.</math> {{RoundBoxBottom}} =Roots of equal absolute value= {{RoundBoxTop|theme=1}} The standard quartic function: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> For <math>x</math> in <math>(1)</math> substitute <math>(p+q).</math> Call this <math>(2).</math> For <math>x</math> in <math>(1)</math> substitute <math>(p-q).</math> Call this <math>(3).</math> Combine <math>(2)</math> and <math>(3)</math> to eliminate <math>q</math> and produce an equation in <math>p:</math> <math>(- 64aaa)pppppp+</math> <math>(- 96aab)ppppp+</math> <math>(- 32aac - 48abb)pppp+</math> <math>(- 32abc - 8bbb)ppp+</math> <math>(+ 16aae - 4abd - 4acc - 8bbc)pp+</math> <math>(+ 8abe - 2bbd - 2bcc)p+</math> <math>(+ add + bbe - bcd)\ =\ 0\ \dots\ (4).</math> We are interested in coefficient <math>0</math> of <math>(4):\ c_0 = add + bbe - bcd.</math> If <math>c_0 == 0,\ p=0</math> is a solution and function <math>(1)</math> has 2 roots of form <math>0 \pm q</math> where <math>q = \sqrt{\frac{-d}{b}}.</math> An example: {{RoundBoxTop|theme=3}} [[File:0320quartic02.png|thumb|400px|'''Graph of quartic function with 2 roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 12x^3 + 31x^2 + 48x -140}{45}.</math> <math>a,b,c,d,e = 1,-12,31,48,-140</math> <math>c_0 = add + bbe - bcd </math><math>= 1(48)(48) + (-12)(-12)(-140) - (-12)(31)(48) = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-48}{-12}} = \sqrt{4} = \pm 2.</math> The 2 roots of equal absolute value are: <math>2, -2.</math> {{RoundBoxBottom}} The method works with complex roots of equal absolute value: {{RoundBoxTop|theme=2}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with 2 complex roots of equal absolute value.''' ]] In the diagram the red line has equation: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}.</math> <math>a,b,c,d,e = 1,-3,-1,-27,-90</math> <math>c_0 = 0.</math> <math>f(x)</math> has roots of equal absolute value. <math>q = \sqrt{\frac{-d}{b}} = \sqrt{\frac{-(-27)}{-3}} = \sqrt{-9} = \pm 3i.</math> The 2 roots of equal absolute value are: <math>3i, -3i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Equal roots= {{RoundBoxTop|theme=2}} Equal roots occur when the function and the slope of the function both equal zero. <math>ax^4 + bx^3 + cx^2 + dx + e = 0\ \dots\ (1)</math> <math>4ax^3 + 3bx^2 + 2cx + d = 0\ \dots\ (2)</math> Begin the process of reducing <math>(1),\ (2)</math> to linear functions. Combine <math>(1),\ (2)</math> to produce 2 cubic functions: <math>Fx^3 + Gx^2 + Hx + J\ \dots\ (1a)</math> where: <math>F = ad;\ G = bd - 4ae;\ H = cd - 3be;\ J = dd - 2ce.</math> <math>fx^3 + gx^2 + hx + j\ \dots\ (2a)</math> where: <math>f = 4a;\ g = 3b;\ h = 2c;\ j = d.</math> Combine <math>(1a),\ (2a)</math> to produce 2 quadratic functions: <math>Kx^2 + Lx + M\ \dots\ (1b)</math> where: <math>K = Gf -Fg;\ L = Hf -Fh;\ M = Jf - Fj.</math> <math>kx^2 + lx + m\ \dots\ (2b)</math> where: <math>k = Fj - Jf;\ l = Gj - Jg;\ m = Hj - Jh.</math> Combine <math>(1b),\ (2b)</math> to produce 2 linear functions: <math>Rx + S\ \dots\ (1c)</math> where: <math>R = Lk - Kl;\ S = Mk - Km.</math> <math>rx + s\ \dots\ (2c)</math> where: <math>r = Km-Mk;\ s = Lm - Ml.</math> From <math>(1c):\ x_1 = \frac{-S}{R}</math> From <math>(2c):\ x_2 = \frac{-s}{r}</math> If <math>x_1 == x_2:</math> <math>\frac{-S}{R} = \frac{-s}{r}</math> <math>Rs = rS</math> <math>Rs - Sr = 0.</math> The value <math>Rs - Sr</math> is in fact: <syntaxhighlight> + 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee - 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd + 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee + 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde + 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee - 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde - 3aabbccdddddd </syntaxhighlight> which, by removing values <math>aa, d</math><math>d</math> (common to all values), may be reduced to: <syntaxhighlight> status = ( + 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee - 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd + 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee + 384abccddde - 54abcddddd + 128acccccee - 80accccdde + 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee - 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde - 3bbccdddd ) </syntaxhighlight> If <math>status == 0,</math> there are at least 2 equal roots which may be calculated as shown below. {{RoundBoxTop|theme=2}} If coefficient <math>d</math> is non-zero, it is not necessary to calculate <math>status.</math> If coefficient <math>d == 0,</math> verify that <math>status = 0</math> before proceeding. {{RoundBoxBottom}} ==No equal roots== {{RoundBoxTop|theme=1}} [[File:0320quartic03.png|thumb|400px|'''Graph of quartic function with no equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 3x^3 - x^2 - 27x - 90}{50}</math> <math>a,b,c,d,e = 1, -3, -1, -27, -90</math> <math>R, S = 15269148, -35977608</math> <math>x_1 = \frac{-S}{R} = \frac{35977608}{15269148} = 2.3562289133617\dots</math> <math>r, s = 35977608, -60634332</math> <math>x_2 = \frac{-s}{r} = \frac{60634332}{35977608} = 1.685335278543253\dots</math> <math>x_1 != x_2.</math> There are no equal roots. {{RoundBoxBottom}} ==Exactly 2 equal roots== {{RoundBoxTop|theme=1}} [[File:0329quartic04.png|thumb|400px|'''Graph of quartic function with exactly 2 equal roots.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 + 6x^3 - 48x^2 - 182x + 735}{100}</math> <math>a,b,c,d,e = 1, 6, -48, -182, 735</math> <math>R, S = -1027353600, -7191475200</math> <math>x_1 = \frac{-S}{R} = \frac{7191475200}{-1027353600} = -7</math> <math>r, s = 7191475200, 50340326400</math> <math>x_2 = \frac{-s}{r} = \frac{-50340326400}{7191475200} = -7</math> <math>x_1 = x_2 = -7.</math> There are 2 equal roots at <math>x = -7.</math> {{RoundBoxTop|theme=2}} The following 3 graphs show the steps that lead to calculation of equal roots at point <math>(-7, 0).</math> <gallery> File:0914quartic00.png|<small>Figure 1. graph of <math>f(x)</math> and 2 associated cubic functions.</small> File:0914quartic01.png|<small>Figure 2. graph of <math>f(x)</math> and 2 associated quadratic functions.</small> File:0914quartic02.png|<small>Figure 3. graph of <math>f(x)</math> and 2 associated linear functions.</small> </gallery> In all graphs, all curves have a common root at point <math>(-7, 0).</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=4}} See [[Quartic_function#Function_as_product_of_linear_function_and_cubic | "Function as product of linear function and cubic"]] above. To calculate all roots: <syntaxhighlight lang=python> # python code. a,b,c,d,e = 1,6,-48,-182,735 # The associated cubic: p = -7 A = a B = A*p + b C = B*p + c D = C*P + d # The associated quadratic: a1 = A b1 = a1*p + B c1 = b1*p + C a1,b1,c1 </syntaxhighlight> <syntaxhighlight> (1, -8, 15) </syntaxhighlight> Roots of quadratic function <math>g(x) = x^2 - 8x + 15</math> are <math>3, 5.</math> All roots of <math>f(x)</math> are <math>-7, -7, 3, 5.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Exactly 3 equal roots== {{RoundBoxTop|theme=1}} [[File:0320_2curves01.png|thumb|400px|'''Graph of quartic function with exactly 3 equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 2x^3 - 36x^2 + 162x - 189}{100}</math> <math>a,b,c,d,e = 1, -2, -36, 162, -189</math> <math>R, S = 0, 0\ \dots\dots\ (1c)</math> <math>r, s = 0, 0\ \dots\dots\ (2c)</math> In this case the calculation of <math>x_1, x_2</math> is not appropriate because there are more than 2 equal roots. Try equations <math>(1b), (2b).</math> Both of these are equivalent to: <math>y = g(x) = x^2 - 6x + 9,</math> blue line in diagram. Discriminant of <math>g(x) = (-6)^2 - 4(1)(9) = 0.\ g(x)</math> has two equal roots at <math>x = \frac{-(-6)}{2(1)} = 3.</math> Therefore <math>f(x)</math> has 3 equal roots at <math>x = 3.</math> {{RoundBoxBottom}} ==Four equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves01.png|thumb|400px|'''Graph of quartic function with 4 equal roots and corresponding cubic.''' </br> <math>g(x) = \frac{f'(x)}{4}.</math> ]] Red line in diagram is of function: <math>y = f(x) = x^4 - 20x^3 + 150x^2 - 500x + 625.</math> <math>a,b,c,d,e = 1, -20, 150, -500, 625</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> <math>K, L, M = 0, 0, 0</math> <math>k, l, m = 0, 0, 0</math> In this case <math>(1b), (2b), (1c), (2c)</math> are all null. This is the only case in which <math>(1b), (2b)</math> are null. <math>(1a), (2a)</math> are both equivalent to: <math>y = g(x) = x^3 - 15x^2 + 75x - 125,</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 4 equal roots at <math>x = 5.</math> {{RoundBoxBottom}} ==Two pairs of equal roots== {{RoundBoxTop|theme=1}} [[File:0321_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal roots and corresponding quadratic.''' ]] Red line in diagram is of function: <math>y = f(x) = \frac{x^4 - 6x^3 - 11x^2 + 60x + 100}{20}.</math> <math>a,b,c,d,e = 1, -6, -11, 60, 100</math> <math>R, S = 0, 0</math> <math>r, s = 0, 0</math> In this case <math>(1c), (2c)</math> are both null. <math>(1b), (2b)</math> are both equivalent to: <math>y = g(x) = \frac{x^2 - 3x - 10}{20},</math> blue line in diagram. <math>g(x)</math> has one root at <math>x = -2</math> and one root at <math>x = 5.</math> Therefore <math>f(x)</math> has 2 equal roots at <math>x = -2</math> and 2 equal roots at <math>x = 5.</math> {{RoundBoxTop|theme=2}} [[File:0408_2curves02.png|thumb|400px|'''Graph of quartic function with 2 pairs of equal, complex roots and corresponding quadratic.''' </br> <math>g(x) = (x-(3-2i))(x-(3+2i)) = x^2 - 6x + 13.</math> </br> <math>f(x) = (g(x))^2.</math> ]] This method is valid for complex roots. For example: <math>y = f(x) = x^4 - 12x^3 + 62x^2 - 156x + 169.</math> <math>a,b,c,d,e = 1,-12,62,-156,169.</math> In this case <math>(1c),\ (2c)</math> are both null. <math>(1b),\ (2b)</math> are both equivalent to: <math>y = g(x) = x^2 - 6x + 13,</math> blue line in diagram. Roots of <math>g(x)</math> are <math>3 \pm 2i.</math> <math>f(x)</math> has 2 roots equal to <math>3 + 2i</math> and 2 roots equal to <math>3 - 2i.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Summmary== {{RoundBoxTop|theme=1}} {| class="wikitable" |- ! || No equal roots !! 2 equal roots !! 3 equal roots !! 4 equal roots !! 2 pairs of equal roots |- | Cubic: 1(a), 2(a) | different | different | different | same | different |- | Quadratic: 1(b), 2(b) | different | different | same, 1root | null | same, 2roots |- | Linear: 1(c), 2(c) | different | same | null | null | null |} {{RoundBoxBottom}} ==Caution== {{RoundBoxTop|theme=4}} [[File:0915quartic02.png|thumb|400px| Calculation of false equal roots.</br> In this example, method calculates 2 legitimate equal roots at <math>(4, 0)</math> and 2 false equal roots at <math>(7.543296089385474,0).</math> <math></math> ]] Black line in diagram has equation: <math>y = f(x) = 0.012684240362811794x^4</math> <math> -\ 0.19522392290249435x^3 </math> <math> +\ 0.7654478458049887x^2 + 0x - 3. </math> <math>f(x)</math> is a quartic function with exactly 2 equal roots and coefficient <math>d</math> missing. Calculation of equal roots of <math>f(x)</math> gives linear functions <math>(1c), (2c)</math> null and quadratic functions <math>(1b), (2b)</math> with equal roots of <math> (4,0), (7.543296089385474,0). </math> Usually, this indicates that <math>f(x)</math> should have 2 equal roots at <math>(4,0)</math> and 2 equal roots at <math>(7.543296089385474,0).</math> It is obvious that <math>7.543296089385474</math> is not a root of <math>f(x).</math> When <math>x = 7.543296089385474,</math> slope of derivative <math>g(x) = 0.</math> Value of <math>f(x)\ != 0.</math> This example indicates that it would be wise to verify that calculated equal roots are in fact valid roots of <math>f(x).</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Depressed quartic= {{RoundBoxTop|theme=1}} A depressed quartic is any quartic function with any one or more of coefficients <math>b,c,d</math> missing. Within this section a depressed quartic has coefficient <math>b</math> missing. To produce the depressed quartic: <math>y = ax^4 + bx^3 + cx^2 + dx + e\ \dots\ (1)</math> <math>y = \frac{(4^4 a^3)(ax^4 + bx^3 + cx^2 + dx + e)}{4^4 a^3}\ \dots\ (2)</math> Let <math>x = \frac{-b + t}{4a}.</math> Substitute in <math>(2),</math> expand and simplify: <math>y = \frac{t^4 + At^2 + Bt + C}{4^4a^3}\ \dots\ (3)</math> where: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> When equated to <math>0,\ (3)</math> becomes the depressed equation: <math>t^4 + At^2 + Bt + C = 0\ \dots\ (4).</math> Be prepared for the possibility that any 1 or more of <math>A, B, C</math> may be zero. ==Coefficient B missing== {{RoundBoxTop|theme=2}} [[File:0322_3curves01.png|thumb|400px|'''Graph of quartic function that resembles a quadratic.''' ]] If coefficient <math>B == 0,\ (4)</math> becomes a quadratic in <math>t^2:</math> <math>t^4 + At^2 + C = 0.</math> <math>(1)</math> has the appearance of a quadratic. The black line: <math>y = f(x) = \frac{x^4 - 4x^3 + 9x^2 - 10x + 5}{10}</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>= 8( 8(-10) - 4(-4)(9) + -64 )</math> <math>= 8(-80 + 144 - 64)</math> <math>= 8(0) = 0.</math> The red line: <math>y' = g(x) = \frac{4x^3 - 12x^2 + 18x - 10}{10}</math> <math>y' = g(x) = 0</math> where <math>x = 1.</math> The grey line: <math>y'' = h(x) = \frac{12x^2 - 24x + 18}{10}</math> * Absolute minima of <math>f(x)</math> and of <math>h(x)</math> and point of inflection of <math>g(x)</math> occur where <math>x = \frac{-b}{4} = 1.</math> * <math>y''</math> is always positive. <math>f(x)</math> is always concave up. * <math>f(1 + p) = f(1-p).</math> {{RoundBoxTop|theme=3}} If <math>(1)</math> contains 2 pairs of equal roots, coefficient <math>B = 0.</math> The converse is not necessarily true. If <math>(1)</math> contains 4 equal roots, coefficients <math>A = B = C = 0.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} ==Coefficient C missing== {{RoundBoxTop|theme=2}} [[File:0501quartic01.png|thumb|400px|'''Graph of quartic function with coefficient C of depressed function missing.''' </br> <math>Y</math> axis compressed for clarity. ]] If coefficient <math>C == 0,\ (4)</math> becomes: <math>t^4 + At^2 + Bt = t(t^3 + At + B) = 0</math> in which case <math>t = 0</math> is a solution and <math>x = \frac{-b}{4a}</math> is a root of <math>(1).</math> Curve (red line) in example has equation: <math>y = f(x) = 8x^4 + 16x^3 + 24x^2 + 89x + 40.</math> Coefficients of depressed function are: <syntaxhighlight lang=python> # python code a,b,c,d,e = 8, 16, 24, 89, 40 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b A,B,C </syntaxhighlight> <syntaxhighlight> (1536, 299008, 0) </syntaxhighlight> Coefficient <math>C</math> of depressed function is missing. <math>t = 0</math> is a solution. Using <math>x = \frac{-b + t}{4a},</math> one root of <math>f(x) = \frac{-16 + 0}{4(8)} = -0.5.</math> {{RoundBoxBottom}} ==Resolvent cubic== {{RoundBoxTop|theme=1}} This section introduces a special cubic function called "resolvent" because it helps to resolve a requirement, the calculation of the roots of the quartic. The depressed quartic: <math>t^4 + At^2 + Bt + C\ \dots\ (1)</math> For <math>t</math> substitute <math>(u+v)\ \dots\ (2)</math> For <math>t</math> substitute <math>(u-v)\ \dots\ (3)</math> <math>(2)+(3):\ 2Auu + 2Avv + 2Bu + 2C + 2uuuu + 12uuvv + 2vvvv\ \dots\ (4)</math> Simplify <math>(4):\ Auu + AV + Bu + C + uuuu + 6uuV + VV\ \dots\ (4a)</math> <math>(2)-(3):\ 4Auv + 2Bv + 8uuuv + 8uvvv\ \dots\ (5)</math> Simplify <math>(5):\ 2Au + B + 4uuu + 4uV\ \dots\ (5a)</math> From <math>(5a):\ 4uV = -(2Au + B + 4uuu)\ \dots\ (5b)</math> <math>(4a)*4u4u:\ 4u4uAuu + A4u(4uV) + 4u4uBu + 4u4uC + 4u4uuuuu + 6uu4u(4uV) + (4uV)(4uV)\ \dots\ (6)</math> In <math>(6)</math> replace <math>4uV</math> with <math>(-(2Au + B + 4uuu)),</math> expand, simplify, gather like terms and result is: <math>Pu^6 + Qu^4 + Ru^2 + S</math> or {{RoundBoxTop|theme=2}} <math>PU^3 + QU^2 + RU + S\ \dots\ (7)</math> where: <math>U = u^2</math> <math>P = 64</math> <math>Q = 32A</math> <math>R = 4A^2 - 16C</math> <math>S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} From <math>(5b):\ V = v^2 = \frac{-(2Au + B + 4uuu)}{4u} = -(\frac{A}{2} + U) - \frac{B}{4u}\ \dots\ (8)</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Some simple changes reduce the number of calculations and also the sizes of coefficients <math>P, Q, R, S.</math> <math>A2 = 8ac - 3b^2</math> where <math>A2 = \frac{A}{2}</math> <math>B4 = 16a^2 d - 8abc + 2b^3</math> where <math>B4 = \frac{B}{4}</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> Then: <math>P = 64</math> <math>Q = 32(A2)(2) = 64A2</math> <math>R = 4(A2)(2)(A2)(2) - 16C = 16A2^2 - 16C</math> <math>S = -(B4)(4)(B4)(4) = -16 B4^2</math> Divide all 4 coefficients by <math>16:</math> <math>P = 4</math> <math>Q = 4A2</math> <math>R = A2^2 - C</math> <math>S = -B4^2</math> <math>V = v^2 = -(A2 + U) - \frac{B4}{u}.</math> A close examination of coefficients <math>R, S</math> shows that both coefficients are always exactly divisible by <math>4.</math> Therefore, all four coefficients may be defined as follows: <math>P = 1</math> <math>Q = A2</math> <math>R = \frac{A2^2 - C}{4}</math> <math>S = \frac{-B4^2}{4}</math> Actual calculations of values <math>R, S</math> are: <syntaxhighlight lang=python> # python code. R = ( (16)*(a)*(a)*(b)*(d) + (16)*(a)*(a)*(c)*(c) + (3)*(b)*(b)*(b)*(b) + (-16)*(a)*(b)*(b)*(c) + (-64)*(a)*(a)*(a)*(e) ) S = ( (64)*(a)*(a)*(a)*(b)*(c)*(d) + (8)*(a)*(b)*(b)*(b)*(b)*(c) + (-16)*(a)*(a)*(b)*(b)*(b)*(d) + (-16)*(a)*(a)*(b)*(b)*(c)*(c) + (-1)*(b)*(b)*(b)*(b)*(b)*(b) + (-64)*(a)*(a)*(a)*(a)*(d)*(d) ) </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} {{RoundBoxBottom}} =Solving quartic equation= This section presents 4 examples that show how to use the depressed quartic and the resolvent cubic to solve the quartic equation. ==Four real roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves01.png|thumb|400px|'''Graphs of quartic function with 4 real roots and associated resolvent cubic.''' </br>Resolvent cubic contains 3 real, positive roots. ]] Calculate roots of: <math>y = f(x) = x^4 - x^3 - 19x^2 - 11x + 30</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,-1,-19,-11,30 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -310 -1320 6669 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 -9920 277696 -1742400 </syntaxhighlight> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = 64x^3 - 9920x^2 + 277696x - 1742400.</math> There are 3 real, positive roots: <math>9, 25, 121.</math> Using 3 roots of <math>g(x),</math> calculate 4 roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code for U in (9, 25, 121) : print ('\nU =', U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** .5 for t in (u+v, u-v) : x = (-b+t) / (4*a) y = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x:',x, '; y:',y) </syntaxhighlight> <syntaxhighlight> U = 9 x: 5.0 ; y: 0.0 x: -3.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 U = 25 x: 5.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -3.0 ; y: 0.0 U = 121 x: 5.0 ; y: 0.0 x: 1.0 ; y: 0.0 x: -2.0 ; y: 0.0 x: -3.0 ; y: 0.0 </syntaxhighlight> Roots of <math>f(x)</math> are: <math>5, 1, -2, -3.</math> {{RoundBoxBottom}} All 3 values of <math>U</math> produce the same results, but not in same sequence. It is not necessary to calculate all 3 roots of resolvent cubic. Any one non-zero root is sufficient to do the job. ==Two real and two complex roots== ===Example 1=== {{RoundBoxTop|theme=4}} [[File:0323_2curves02.png|thumb|400px|'''Graphs of quartic function with 2 real and 2 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 + 2x^3 + 18x^2 - 70x - 87</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 1,2,18,-70,-87 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> 264 -5568 -12208 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 64 8448 474112 -31002624 </syntaxhighlight> {{RoundBoxBottom}} Calculate one real root of cubic function: <math>y = g(x) = 64x^3 + 8448x^2 + 474112x -31002624.</math> <math>36</math> is one real root. Choose <math>U = 36.</math> Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 36 u1 = U**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) if V >= 0 : v = V**.5 else : v = 1j * (-V)**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =',x, '; sum =',sum ) </syntaxhighlight> <syntaxhighlight> x = 3.0 ; sum = 0.0 x = -1.0 ; sum = 0.0 x = (-2+5j) ; sum = 0j x = (-2-5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=4}} [[File:0321_2curves00.png|thumb|400px| Coefficient <math>d</math> of <math>g(x) = 0.</math> <math></math> ]] Calculate roots of: <math>y = f(x)</math> <math>= 3x^4 - 6x^3 - 41x^2 +44x - 189</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> a,b,c,d,e = 3, -6, -41, 44, -189 A = 16*a*c - 6*b*b B = 64*a*a*d - 32*a*b*c + 8*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b print (A,B,C) </syntaxhighlight> <syntaxhighlight> -2184, 0, -1229040 </syntaxhighlight> Notice that coefficient <math>B = 0.</math> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> P = 64 Q = 32*A R = 4*A*A - 16*C S = -B*B print (P,Q,R,S) </syntaxhighlight> <syntaxhighlight> 1, -1092, 605376, 0 </syntaxhighlight> Notice that coefficient <math>S = 0.</math> {{RoundBoxBottom}} Calculate roots of cubic function: <math>y = g(x) = x^3 - 1092x^2 + 605376x + 0.</math> Roots are <math>0, 546 \pm 554.3103823671355j.</math> Value <math>0</math> cannot be used because it will cause error <code>Divide by zero</code> at statement <code>V = -(A/2 + U) - B/(4*u)</code>. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = 546+554.3103823671355j print ('\nU =',U) sqrtU = U ** 0.5 for u in (sqrtU, -sqrtU) : V = -(A/2 + U) - B/(4*u) v = V ** 0.5 s1 = '\nu,v' print (s1,eval(s1)) for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> U = (546+554.3103823671355j) u,v ((25.729935131257832+10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (4.788322521876306 + 0j) ; sum = (1.9895196601282805e-13 + 0j) x = (0.5 + 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) u,v ((-25.729935131257832-10.771701901683684j), (25.729935131257832-10.771701901683684j)) x = (0.5 - 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j) x = (-3.7883225218763052 + 0j) ; sum = (1.7053025658242404e-13 + 0j) </syntaxhighlight> Values of <math>x</math> are: <math>-3.7883225218763052, 4.788322521876306, 0.5 \pm 1.795283650280614j</math> <math></math><math></math> {{RoundBoxBottom}} ====Depressed quartic as quadratic==== In this example coefficient <math>B</math> of depressed quartic <math>= 0.</math> Therefore, resolvent cubic can be ignored and depressed quartic processed as quadratic in <math>T = t^2.</math> <math>t^4 - 2184t^2 + (0)t - 1229040</math> <math>T^2 - 2184T - 1229040</math> where <math>T = t^2.</math> Solutions of this quadratic are: <math>T_1, T_2 = 2648.1182474349434, -464.11824743494344</math> <syntaxhighlight lang=python> T1,T2 = 2648.1182474349434, -464.11824743494344 t1 = T1 ** 0.5; t2 = ((-T2) ** 0.5) * 1j for t in (t1,-t1,t2,-t2) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> x = 4.788322521876305 ; sum = -1.7053025658242404e-13 x = -3.788322521876305 ; sum = -1.7053025658242404e-13 x = (0.5 + 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) x = (0.5 - 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j) </syntaxhighlight> or <syntaxhighlight> x = 0.5 ± 4.288322521876305, 0.5 ± 1.7952836502806138j </syntaxhighlight> With precision of 15, values of <math>x</math> are same as those shown above. When roots of quartic function are of form <code>p ± q, p ± r,</code> coefficient <math>B</math> of depressed function <math>= 0.</math> ==Four complex roots== {{RoundBoxTop|theme=4}} [[File:0323_2curves03.png|thumb|400px|'''Graphs of quartic function with 4 complex roots and associated resolvent cubic.''']] Calculate roots of: <math>y = f(x) = x^4 - 20x^3 + 408x^2 + 2296x + 18020</math> {{RoundBoxBottom}} Calculate coefficients of depressed quartic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 4128 344064 9683200 </syntaxhighlight> {{RoundBoxBottom}} Calculate coefficients of resolvent cubic: {{RoundBoxTop|theme=2}} <syntaxhighlight> 64 132096 -86769664 -118380036096 </syntaxhighlight> {{RoundBoxBottom}} Calculate one root of cubic function: <math>y = g(x) = 64x^3 + 132096x^2 - 86769664x - 118380036096.</math> There are 3 real roots: <math>-2304, -784, 1024.</math> Choose <math>U = -784.</math> Negative <math>U</math> is chosen here to show that any 1 of the roots produces the correct result. Calculate roots of <math>f(x):</math> {{RoundBoxTop|theme=2}} <syntaxhighlight lang=python> # python code U = -784 u1 = 1j * (-U)**.5 for u in (u1, -u1) : V = -(A/2 + U) - B/(4*u) v = V**.5 for t in (u+v, u-v) : x = (-b+t)/(4*a) # Check result. Expecting sum = 0. sum = a*x**4 + b*x**3 + c*x**2 + d*x + e print ('x =', x,'; sum =',sum) </syntaxhighlight> <syntaxhighlight> # python expresses complex numbers with 'j'. x = (13+19j) ; sum = 0j x = (-3-5j) ; sum = 0j x = (13-19j) ; sum = 0j x = (-3+5j) ; sum = 0j </syntaxhighlight> {{RoundBoxBottom}} =Quartic formula= The substitutions made above can be used to produce a formula for the solution of the quartic equation. See main articles [https://en.wikipedia.org/wiki/Quartic_equation#The_general_case "The general case" ] or [https://en.wikipedia.org/wiki/Quartic_function#General_formula_for_roots "General formula for roots."] {{RoundBoxTop|theme=2}} Both links above point to formula for equation <math>x^4 + ax^3 + bx^2 + cx + d = 0.</math> {{RoundBoxBottom}} Given quartic equation: <math>ax^4 + bx^3 + cx^2 + dx + e = 0,</math> calculate the 4 values of <math>x.</math> <math>x = \frac{-b + t}{4a}</math> where: {{RoundBoxTop|theme=2}} Coefficients of depressed quartic: <math>A = 16ac - 6b^2</math> <math>B = 64a^2 d - 32abc + 8b^3</math> <math>C = 256a^3e - 64a^2 bd + 16ab^2 c - 3 b^4</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of resolvent cubic: <math>a_1 = P = 64</math> <math>b_1 = Q = 32A</math> <math>c_1 = R = 4A^2 - 16C</math> <math>d_1 = S = -B^2</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} Coefficients of depressed cubic: <math>A_1 = 9a_1 c_1 - 3b_1^2</math> <math>B_1 = 27a_1^2d_1 - 9a_1b_1c_1 + 2b_1^3</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of resolvent cubic: <math>C_1 = \frac{-A_1}{3} = b_1^2 - 3a_1 c_1</math> <math>\Delta = B_1^2 - 4C_1^3\ \dots\dots\ \Delta</math> may be negative. <math>\delta = \sqrt{\Delta}</math> <math>W = \frac{-B_1 + \delta}{2}</math> <math>w = \sqrt[3]{W}</math> <math>t_1 = w + \frac{C_1}{w}</math> <math>U = \frac{-b_1 + t_1}{3a_1}</math> {{RoundBoxBottom}} {{RoundBoxTop|theme=2}} One root of quartic: <math>u = \sqrt{U}\ \dots\dots\ u</math> may be positive or negative. <math>V = -(\frac{A}{2} + U) - \frac{B}{4u}</math> <math>v = \sqrt{V}\ \dots\dots\ v</math> may be positive or negative. <math>t = u + v</math> {{RoundBoxBottom}} Formula above produces one value of <math>x.</math> Python code below utilizes <math>\pm \sqrt{U}</math> and <math>\pm \sqrt{V}</math> to produce 4 values of <math>t</math> and then, four values of <math>x.</math> ==An example:== {{RoundBoxTop|theme=4}} [[File:0330quartic01.png|thumb|400px|'''Graph of quartic function with 2 real roots.'''</br><math>Y</math> axis compressed for clarity.]] Calculate roots of <math>f(x) = 4x^4 + 4x^3 - 75x^2 - 776x - 1869.</math> <syntaxhighlight lang=python> # Python code. a,b,c,d,e = 4, 4, -75, -776, -1869 values_of_t = [ t # Coefficients of depressed quartic: for A in (16*a*c - 6*b*b,) for B in (64*a*a*d -32*a*b*c + 8*b*b*b,) for C in (256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b,) # Coefficients of resolvent cubic: for a1 in (64,) for b1 in (32*A,) for c1 in (4*A*A - 16*C,) for d1 in (-B*B,) for U in [ # The resolvent cubic: (-b1+t1)/(3*a1) # Coefficients of depressed resolvent cubic: for A1 in (9*a1*c1 - 3*b1*b1,) for B1 in (27*a1*a1*d1 - 9*a1*b1*c1 + 2*b1*b1*b1,) # One root of resolvent cubic: for C1 in (-A1/3,) for Δ in (B1*B1 - 4*C1*C1*C1,) for δ in (Δ**0.5,) for W in ((-B1 + δ)/2,) for w in (W**(1/3),) for t1 in (w + C1/w,) # See note below. ] # Prepare to calculate 4 values of t. for u1 in (U**.5,) for v1 in ( -(A/2 + U) ,) # Calculate 4 values of t. for u in (u1, -u1,) for V in ( v1 - B/(4*u),) for v in (V**.5,) for t in (u+v, u-v) ] print ('values_of_t =', values_of_t) </syntaxhighlight> {{RoundBoxTop|theme=4}} <syntaxhighlight> values_of_t = [116, -44, (-36+64j), (-36-64j)] </syntaxhighlight> Because <math>f(t)</math> is a depressed quartic function, sum of four <code>values_of_t</code> <math>= 116 - 44 - 36(2) = 0.</math> {{RoundBoxBottom}} <syntaxhighlight lang=python> # Python code. # Calculate 4 separate roots. values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] print ('values_of_x =', values_of_x) </syntaxhighlight> <syntaxhighlight lang=python> values_of_x = [7, -3, (-2.5 + 4j), (-2.5 - 4j)] </syntaxhighlight> In python the imaginary part of a complex number is shown with <math>j</math> instead of <math>i.</math> If <math>A == B == C == 0,\ f(x)</math> contains 4 equal roots and <math>x = \frac{-b}{4a}.</math> If <math>f(x)</math> contains 3 or more equal roots, statement <code>for t1 in (w + C1/w,)</code> fails with divisor <math>w = 0.</math> Before using this formula, check for equal roots as in [[Quartic_function#Exactly_3_equal_roots | "Exactly 3 equal roots"]] above. {{RoundBoxTop|theme=1}} Values displayed above have been edited slightly. Actual calculated values were: <syntaxhighlight> values_of_x = [7.000000000000001, -3.0000000000000044, (-2.499999999999998+4.000000000000001j), (-2.4999999999999987-4.000000000000001j)] </syntaxhighlight> {{RoundBoxBottom}} {{RoundBoxBottom}} ==In practice== {{RoundBoxTop|theme=5}} The following Python code implements the quartic formula. However, under statement <code>if B4 == 0 :</code> there is code that processes the depressed quartic as a quadratic in <math>T = t^2.</math> This ensures that execution of formula does not fail with error <code>Divide by zero</code> at statement <code>for V in ( v1 - B4/u,).</code> {{RoundBoxTop|theme=4}} <syntaxhighlight lang=python> # python code import cmath cxSqrt = cmath.sqrt # Square root of complex number. def rootsOfQuartic (abcde) : ''' x1,x2,x3,x4 = rootsOfQuartic ((a,b,c,d,e)) Each member of input must be int or float or Decimal object. Int or Decimal object in input is quietly converted to float. Output may be None. ''' def formatResults (x1x2x3x4) : ''' This function improves appearance of results. (8 + 0j) becomes 8.0 ''' values_of_x = list (x1x2x3x4) for p in (0,1,2,3) : v = values_of_x[p] if isinstance (v, complex) and (v.imag == 0) : values_of_x[p] = v.real return values_of_x status = 0 try : a,b,c,d,e = [ float(v) for v in abcde ] except : status = 1 if status : print ('rootsOfQuartic () 1: Error creating coefficients a,b,c,d,e.') return None if a == 0 : print ('rootsOfQuartic () 2: Coefficient a must be non-zero.') return None # Coefficients of depressed quartic, modified. # A = 16*a*c - 6*b*b A2 = 8*a*c - 3*b*b # B = 64*a*a*d - 32*a*b*c + 8*b*b*b B4 = 16*a*a*d - 8*a*b*c + 2*b*b*b C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b if B4 == 0 : # B = 0. # Result returned from this section is type tuple, indicating that coefficient B4 = 0. if A2==C==0 : # 4 equal roots. root = -b/(4*a) return tuple(formatResults((root,root,root,root))) # t**4 + At**2 + (0)t + C # Depressed quartic is quadratic in T: # T**2 + AT + C where T = t**2 # T**2 + 2(A2)T + C where A = 2(A2) # -2(A2) +/- (4(A2)(A2) - 4C)**0.5 # T = -------------------------------- = -A2 +/- ((A2)(A2) - C)**0.5 # 2 disc = A2*A2 - C if disc >= 0 : root = disc ** 0.5 else : root = ((-disc) ** 0.5) * 1j T1 = -A2 - root ; T2 = -A2 + root t1 = cxSqrt(T1) ; t2 = cxSqrt(T2) values_of_t = (t1,-t1,t2,-t2) values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] return tuple(formatResults(values_of_x)) # B4 is non-zero. Therefore, all of (S, U, u) are non-zero. P,Q,R,S = 1, A2, (A2*A2 - C)/4, -B4*B4/4 # str1 = 'P,Q,R,S' ; print (str1, eval(str1)) U = oneRootOfCubic((P,Q,R,S)) # Resolvent cubic. if U > 0 : sqrtU = U ** 0.5 elif U == 0 : # This should not happen. print ('rootsOfQuartic () 3: Internal error.') return None else : sqrtU = ((-U) ** 0.5) * 1j v1 = -(A2+U) values_of_t = [ t for u in (sqrtU, -sqrtU) for V in ( v1 - B4/u, ) for v in ( cxSqrt(V), ) for t in (u+v, u-v) ] values_of_x = [ (-b + t)/(4*a) for t in values_of_t ] # Result returned from this section is type list, indicating that coefficient B4 != 0. return formatResults(values_of_x) </syntaxhighlight> For function <code>oneRootOfCubic()</code> see [[Cubic_function#In_practice | Cubic_function: In_practice.]] {{RoundBoxBottom}} ===Examples=== {{RoundBoxTop|theme=2}} Python function <code>equalRoots()</code> below implements <code>status</code> as presented under [https://en.wikiversity.org/wiki/Quartic_function#Equal_roots Equal roots] above. <syntaxhighlight lang=python> # python code def equalRoots(abcde) : ''' This function returns True if quartic function contains at least 2 equal roots. ''' a,b,c,d,e = abcde aa = a*a ; aaa = aa*a bb = b*b ; bbb = bb*b ; bbbb = bb*bb cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd ee = e*e ; eee = ee*e ; eeee = ee*ee v1 = ( +2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd +1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee +12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd +20*bb*ccc*dd*e ) v2 = ( -768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e -480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e -216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd ) return (v1+v2) == 0 t1 = ( ((1, -1, -19, -11, 30), '4 unique, real roots.'), ((4, 4,-119, -60, 675), '4 unique, real roots, B4 = 0.'), ((1, 6, -48,-182, 735), '2 equal roots.'), ((1,-12, 50, -84, 45), '2 equal roots. B4 = 0.'), ((1,-20, 146,-476, 637), '2 equal roots, 2 complex roots.'), ((1,-12, 58,-132, 117), '2 equal roots, 2 complex roots. B4 = 0.'), ((1, -2, -36, 162, -189), '3 equal roots.'), ((1,-20, 150,-500, 625), '4 equal roots. B4 = 0.'), ((1, -6, -11, 60, 100), '2 pairs of equal roots, B4 = 0.'), ((4, 4, -75,-776,-1869), '2 complex roots.'), ((1,-12, 33, 18, -208), '2 complex roots, B4 = 0.'), ((1,-20, 408,2296,18020), '4 complex roots.'), ((1,-12, 83, -282, 442), '4 complex roots, B4 = 0.'), ((1,-12, 62,-156, 169), '2 pairs of equal complex roots, B4 = 0.'), ) for (abcde, comment) in t1 : print () fourRoots = rootsOfQuartic (abcde) print (comment) print (' Coefficients =', abcde) print (' Four roots =', fourRoots) print (' Equal roots detected:', equalRoots(abcde)) # Check results. a,b,c,d,e = abcde for x in fourRoots : # To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0 sum = (a*x**4 + b*x**3 + c*x**2 + d*x + e) if sum : # Create exception. 1/0 </syntaxhighlight> <syntaxhighlight> 4 unique, real roots. Coefficients = (1, -1, -19, -11, 30) Four roots = [5.0, 1.0, -2.0, -3.0] Equal roots detected: False 4 unique, real roots, B4 = 0. Coefficients = (4, 4, -119, -60, 675) Four roots = (2.5, -3.0, 4.5, -5.0) Equal roots detected: False 2 equal roots. Coefficients = (1, 6, -48, -182, 735) Four roots = [5.0, 3.0, -7.0, -7.0] Equal roots detected: True 2 equal roots. B4 = 0. Coefficients = (1, -12, 50, -84, 45) Four roots = (3.0, 3.0, 5.0, 1.0) Equal roots detected: True 2 equal roots, 2 complex roots. Coefficients = (1, -20, 146, -476, 637) Four roots = [7.0, 7.0, (3+2j), (3-2j)] Equal roots detected: True 2 equal roots, 2 complex roots. B4 = 0. Coefficients = (1, -12, 58, -132, 117) Four roots = ((3+2j), (3-2j), 3.0, 3.0) Equal roots detected: True 3 equal roots. Coefficients = (1, -2, -36, 162, -189) Four roots = [3.0, 3.0, 3.0, -7.0] Equal roots detected: True 4 equal roots. B4 = 0. Coefficients = (1, -20, 150, -500, 625) Four roots = (5.0, 5.0, 5.0, 5.0) Equal roots detected: True 2 pairs of equal roots, B4 = 0. Coefficients = (1, -6, -11, 60, 100) Four roots = (5.0, -2.0, 5.0, -2.0) Equal roots detected: True 2 complex roots. Coefficients = (4, 4, -75, -776, -1869) Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)] Equal roots detected: False 2 complex roots, B4 = 0. Coefficients = (1, -12, 33, 18, -208) Four roots = ((3+2j), (3-2j), 8.0, -2.0) Equal roots detected: False 4 complex roots. Coefficients = (1, -20, 408, 2296, 18020) Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)] Equal roots detected: False 4 complex roots, B4 = 0. Coefficients = (1, -12, 83, -282, 442) Four roots = ((3+5j), (3-5j), (3+2j), (3-2j)) Equal roots detected: False 2 pairs of equal complex roots, B4 = 0. Coefficients = (1, -12, 62, -156, 169) Four roots = ((3+2j), (3-2j), (3+2j), (3-2j)) Equal roots detected: True </syntaxhighlight> When description contains note <math>B4 = 0,</math> depressed quartic was processed as quadratic in <math>t^2.</math> {{RoundBoxBottom}} {{RoundBoxBottom}} =Two Conic Sections= Examples of conic sections include: ellipse, circle, parabola and hyperbola. This section presents examples of two conic sections, circle and ellipse, and how to calculate the coordinates of the point/s of intersection, if any, of the two sections. Let one section with name <math>ABCDEF</math> have equation <math>Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.</math> Let other section with name <math>abcdef</math> have equation <math>ax^2 + by^2 + cxy + dx + ey + f = 0.</math> Because there can be as many as 4 points of intersection, a special "resolvent" quartic function is used to calculate the <math>x</math> coordinates of the point/s of intersection. Coefficients of associated "resolvent" quartic are calculated as follows: <syntaxhighlight lang=python> # python code def intersection_of_2_conic_sections (abcdef, ABCDEF) : ''' A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF) where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function: y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_ ''' A,B,C,D,E,F = ABCDEF a,b,c,d,e,f = abcdef G = ((-1)*(B)*(a) + (1)*(A)*(b)) H = ((-1)*(B)*(d) + (1)*(D)*(b)) I = ((-1)*(B)*(f) + (1)*(F)*(b)) J = ((-1)*(C)*(a) + (1)*(A)*(c)) K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c)) L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c)) M = ((-1)*(E)*(f) + (1)*(F)*(e)) g = ((-1)*(C)*(b) + (1)*(B)*(c)) h = ((-1)*(E)*(b) + (1)*(B)*(e)) A_ = (J)*(g) + (G)*(G) B_ = (J)*(h) + (K)*(g) + (2)*(G)*(H) C_ = (K)*(h) + (L)*(g) + (2)*(G)*(I) + (H)*(H) D_ = (L)*(h) + (M)*(g) + (2)*(H)*(I) E_ = (M)*(h) + (I)*(I) str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_) print (str1) return A_,B_,C_,D_,E_ </syntaxhighlight> == With no common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections06.png|thumb|400px|'''Two conic sections with no common point.''' </br> Resolvent quartic function (black curve) has no real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 80 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.98x^2 + 6850.532x - 22836.7009.</math> <math>f(x)</math> has no real roots. Therefore, there is no point of intersection. {{RoundBoxBottom}} ==With one common point== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections05.png|thumb|400px|'''Two conic sections with one common point.''' </br> Resolvent quartic function (black curve) has two equal, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 73 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.84x^2 + 7456.48x - 24355.36.</math> Roots of <math>f(x)</math> are: <math>(-14-22.978250586152114j),</math> <math>(-14+22.978250586152114j),</math> <math>5.8, 5.8.</math> <math>f(x)</math> has 2 equal, real roots at <math>x = 5.8,</math> effectively 1 real root where <math>x = 5.8</math> Therefore, there is one point of intersection where <math>x = 5.8.</math> {{RoundBoxBottom}} ==With two common points== ===Example 1=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections04.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two unique, real roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 6.8x - 17.6y + 64 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 - 16.4x^3 - 432.66x^2 + 8235.556x - 26681.1841.</math> Roots of <math>f(x)</math> are: <math>(-14.361578825892241-23.341853011785357j),</math> <math>(-14.361578825892241+23.341853011785357j),</math> <math>4.59885619413921, 7.72430145764527.</math> <math>f(x)</math> has 2 unique, real roots at <math>x = 4.59885619413921, 7.72430145764527.</math> Therefore, there are two points of intersections where <math>x = 4.59885619413921, 7.72430145764527.</math> {{RoundBoxBottom}} ===Example 2=== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections01.png|thumb|400px|'''Two conic sections with two common points.''' </br> Resolvent quartic function (black curve) has two pairs of equal roots.</br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 18.8x - 1.6y + 53 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = -x^4 + 37.6x^3 - 504.24x^2 + 2835.04x - 5685.16.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 13, 13.</math> <math>f(x)</math> has 2 pairs of equal roots at <math>x = 5.8, 13,</math> effectively 2 real roots. Therefore, there are two points of intersection where <math>x = 5.8, 13.</math> {{RoundBoxBottom}} ==With 3 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections03.png|thumb|400px|'''Two conic sections with three common points.''' </br> Resolvent quartic function (black curve) has one pair of equal roots and 2 unique, real roots. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 17.6x - 3.2y + 55 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x) = x^4 - 32.2x^3 + 366.69x^2 - 1784.428x + 3165.1876.</math> Roots of <math>f(x)</math> are:<math>5.8, 5.8, 6.83589838486224, 13.7641016151377 .</math> <math>f(x)</math> has 1 pair of equal roots at <math>x = 5.8</math> and 2 unique, real roots at <math>x = 6.83589838486224, 13.7641016151377,</math> effectively 3 real roots. Therefore, there are three points of intersection where <math>x = 5.8, 6.83589838486224, 13.7641016151377.</math> {{RoundBoxBottom}} ==With 4 common points== {{RoundBoxTop|theme=2}} [[File:0308_2conic_sections02.png|thumb|400px|'''Two conic sections with four common points.''' </br> Resolvent quartic function (black curve) has 4 unique, real roots. </br> <math>y</math> axis of quartic function is compressed to illustrate shape of curve. ]] Let ellipse (red curve) have equation: <math>1.89x^2 + 1.61y^2 + 0.96xy - 36.3x - 11.6y + 130.25 = 0.</math> Let circle (blue curve) have equation: <math>x^2 + y^2 - 18.8x - 1.6y + 62.99 = 0.</math> Then, resolvent quartic function (black curve) has equation: <math>y = f(x)</math> <math>= -x^4 + 37.6x^3 - 504.4398x^2 + 2838.79624x - 5544.61147921.</math> Roots of <math>f(x)</math> are:<math>4.36661032156283,</math><math>8.77936456353008,</math> <math>10.0206354364699,</math><math>14.4333896784371.</math> <math>f(x)</math> has 4 real roots as shown above. Therefore, there are four points of intersection where <math>x = 4.36661032156283,</math> <math>8.77936456353008,</math><math>10.0206354364699,</math><math>14.4333896784371.</math> {{RoundBoxBottom}} ===Values of y=== Section above calculates <math>x</math> coordinates of the four common points. This section calculates <math>y</math> coordinates. Here is the lazy way to do it: {{RoundBoxTop|theme=2}} [[File:0818ResQuartic.png|thumb|400px|'''Graphs of 2 curves, red and blue, with coefficients of <math>x,y</math> reversed.''' </br> Roots of "resolvent" quartic, black curve, are interpreted as values of <math>y.</math> ]] <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 # Coefficients of x,y are reversed. abcde = intersection_of_2_conic_sections ((b,a,0,e,d,f), (B,A,C,E,D,F)) values_of_y = rootsOfQuartic (abcde) print (values_of_y) </syntaxhighlight> <syntaxhighlight> [1.6215767432213488, -0.021576743221348723, 5.862095579401657, -4.262095579401658] </syntaxhighlight> By inspection of the diagram above, it is obvious that: <syntaxhighlight lang=python> # python code four_common_points = ( (4.36661032156283, 1.6215767432213488), (8.77936456353009, -4.262095579401658), (10.0206354364699, 5.862095579401657), (14.4333896784372, -0.021576743221348723), ) </syntaxhighlight> Check results: <syntaxhighlight lang=python> # python code def sum_zero (input) : ''' This function calculates sum of all values in input. If sum is very close to 0, this function returns 0. ''' sum = 0 for v in input : sum += v if sum == 0 : return sum tolerance = 1e-14 max = sorted([ abs(v) for v in input ])[-1] if abs(sum) < tolerance*max : return 0 return sum # The ellipse: A,B,C,D,E,F = 1.89, 1.61, 0.96, -36.3, -11.6, 130.25 # The circle: a,b,c,d,e,f = 1, 1, 0, -18.8, -1.6, 62.99 for (x,y) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum1 = sum_zero(values) if sum1 : print ('Error: sum1 =',sum1) values = a*x**2, b*y**2, d*x, e*y, f sum2 = sum_zero(values) if sum2 : print ('Error: sum2 =',sum2) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. Check results with coefficients of <math>x, y</math> reversed: <syntaxhighlight lang=python> # python code # The ellipse: A,B,C,D,E,F = B,A,C,E,D,F # The circle: a,b,c,d,e,f = b,a,c,e,d,f for (y,x) in four_common_points : values = A*x**2, B*y**2, C*x*y, D*x, E*y, F sum3 = sum_zero(values) if sum3 : print ('Error: sum3 =',sum3) values = a*x**2, b*y**2, d*x, e*y, f sum4 = sum_zero(values) if sum4 : print ('Error: sum4 =',sum4) </syntaxhighlight> With tolerance of <math>1e-14</math> there are no errors. {{RoundBoxBottom}} =Links to related topics= {{RoundBoxTop|theme=1}} [[Cubic_function#Cubic_formula | "Cubic formula"]] [[Complex_square_root | "Complex square root" ]] {{RoundBoxBottom}} [[Category:Mathematics]] q477h9ad6qywadk5qns9ru2z6aebq2u 4 285491 2843988 2842644 2026-11-02T16:01:45Z Alexis Jazz 791434 Updating gadget usage statistics from [[Special:GadgetUsage]] ([[phab:T121049]]) 2843988 wikitext text/x-wiki {{#ifexist:Project:GUS2Wiki/top|{{/top}}|This page provides a historical record of [[Special:GadgetUsage]] through its page history. To get the data in CSV format, see wikitext. To customize this message or add categories, create [[/top]].}} The following data is cached, and was last updated 2026-11-01T06:53:07Z. A maximum of {{PLURAL:5000|one result is|5000 results are}} available in the cache. {| class="sortable wikitable" ! Gadget !! data-sort-type="number" | Number of users !! data-sort-type="number" | Active users |- |CleanDeletions || 75 || 0 |- |EnhancedTalk || 1403 || 3 |- |HideFundraisingNotice || 841 || 10 |- |HotCat || 924 || 13 |- |LintHint || 121 || 3 |- |Round Corners || 1192 || 3 |- |contribsrange || 389 || 4 |- |dark-mode || 138 || 1 |- |dark-mode-toggle || 193 || 4 |- |edittop || 522 || 8 |- |popups || 886 || 4 |- |purge || 740 || 10 |- |sidebartranslate || 565 || 2 |- |usurper-count || 120 || 1 |} * [[Special:GadgetUsage]] * [[m:Meta:GUS2Wiki/Script|GUS2Wiki]] <!-- data in CSV format: CleanDeletions,75,0 EnhancedTalk,1403,3 HideFundraisingNotice,841,10 HotCat,924,13 LintHint,121,3 Round Corners,1192,3 contribsrange,389,4 dark-mode,138,1 dark-mode-toggle,193,4 edittop,522,8 popups,886,4 purge,740,10 sidebartranslate,565,2 usurper-count,120,1 --> ts0ihvb13zpxb10lfgvya75pzrlhhb9 2 289273 2844008 2843942 2026-11-02T19:59:06Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844008 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... The geometry of the atom is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. S discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a compound series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one important exception to this progression which will concern us later). The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The 4-simplex is a configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is a 4-simplex of a distinct radius. The largest regular convex 4-polytope has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 6p66l0192qrp0y6cm7znstgga7o1w4l 2844009 2844008 2026-11-02T19:59:20Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844009 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... The geometry of the atom is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. S discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a compound series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one important exception to this progression which will concern us later). The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The 4-simplex is a configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is a 4-simplex of a distinct radius. The largest regular convex 4-polytope has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} ok75r3nq9iasatpr54kalqf4g1ikg7r 2844010 2844009 2026-11-02T20:00:29Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844010 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... Describing the geometry of the atom is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. S discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a compound series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one important exception to this progression which will concern us later). The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The 4-simplex is a configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is a 4-simplex of a distinct radius. The largest regular convex 4-polytope has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} ewf38shp07v3v4t8prldiohjglmjo0p 2844011 2844010 2026-11-02T20:11:42Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844011 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... Describing the geometry of the atom is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a compound series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one important exception to this progression which will concern us later). The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is a 4-simplex of a distinct radius. The largest regular convex 4-polytope has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope nest within it. It is comprehensive in that it contains instances of every regular convex polytope which can be constructed in Euclidean 4-space. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} ovjj2cp4z4m35k6z5f5kjtafs3xbvis 2844018 2844011 2026-11-02T20:43:01Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844018 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the geometry of the regular convex 4-polytopes is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one important exception to this progression which will concern us later). The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is a 4-simplex of a distinct radius. The largest regular convex 4-polytope has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope nest within it. It is comprehensive in that it contains instances of every regular convex polytope which can be constructed in Euclidean 4-space. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} iqn2ijt75m458ec94eur94dvxgls0cr 2844019 2844018 2026-11-02T20:43:34Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844019 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the geometry of the regular convex 4-polytopes is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one important exception to this progression which will concern us later). The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is a 4-simplex of a distinct radius. The largest regular convex 4-polytope has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope nest within it. It is comprehensive in that it contains instances of every regular convex polytope which can be constructed in Euclidean 4-space. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 9h7uejodg50grmc0pdhsb374az0fjxv 2844022 2844019 2026-11-02T21:13:40Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844022 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the geometry of the regular convex 4-polytopes is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one important exception to this progression which will concern us later). The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex of a distinct radius. The largest regular convex 4-polytope is called the 120-cell. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can be constructed in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are 120 possible atomic elements. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} jfw88xdf6tw0oi7f09vhfgwdwjltam4 2844025 2844022 2026-11-02T21:17:04Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844025 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the geometry of the regular convex 4-polytopes is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex of a distinct radius. The largest regular convex 4-polytope is called the 120-cell. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can be constructed in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are 120 possible atomic elements. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} jv004p4vuq8eg0jz0q1qwwvf08klco6 2844026 2844025 2026-11-02T21:18:38Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844026 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the geometry of the regular convex 4-polytopes is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex. The largest regular convex 4-polytope is called the 120-cell. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can be constructed in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are 120 possible atomic elements. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 9xosxslrh45fkvpu2kmhugh0pfegk78 2844027 2844026 2026-11-02T21:20:31Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844027 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the geometry of the regular convex 4-polytopes is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. The smallest regular convex 4-polytope is the 4-simplex. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex. The largest regular convex 4-polytope is called the 120-cell. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are 120 possible atomic elements. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} pt3vuno44pbw9sptf0sc939njg6i5la 2844029 2844027 2026-11-02T21:32:29Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844029 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the [[Polyscheme|geometry of the regular convex 4-polytopes]] is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} jcumuvsu4iwhykpcuwqcm8j3e9n3a5c 2844031 2844029 2026-11-02T21:51:51Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844031 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the [[Polyscheme|geometry of the regular convex 4-polytopes]] is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-polytope, nor to any other 4-polytope in the series except the largest, the 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-polytopes. Thus there are two incompatible smallest building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 1cf6lwsmes3v0l40k91oskv0wiyh1sj 2844032 2844031 2026-11-02T21:53:42Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844032 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the [[Polyscheme|geometry of the regular convex 4-polytopes]] is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two incompatible smallest building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 6391jp547wuys5sw1xe4glt67kxok12 2844034 2844032 2026-11-02T22:04:55Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844034 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is that of the six regular convex 4-polytopes. Describing the [[Polyscheme|geometry of the regular convex 4-polytopes]] is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytopes in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the latter will build the other three 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} l49iv0mnvpu6e3x4e1nsz4hpun5vlq2 2844035 2844034 2026-11-02T22:07:07Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844035 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is compounded from that of the six regular convex 4-polytopes. Describing the [[Polyscheme|geometry of the regular convex 4-polytopes]] is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytopes in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the latter will build the other three 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} let0ltcdi182arpcj7h1ptnacwwujsx 2844036 2844035 2026-11-02T22:08:06Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844036 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is compounded from that of the six regular convex 4-polytopes. Describing the [[Polyscheme|geometry of the regular convex 4-polytopes]] is beyond the scope of this paper, but we may start with the proton, which we identify with the symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The geometry of the proton is that of a 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the latter will build the other three 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 3x6gjim5imyj392b7chlzmx73amxzi5 2844039 2844036 2026-11-02T22:14:30Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844039 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their symmetry is compounded from that of the six regular convex 4-polytopes. Describing the rotational geometry of the [[Polyscheme|regular convex 4-polytopes]] is beyond the scope of this paper, but we may start with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the latter will build the other three 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 00w7phdk4rry24gca5g173j2oibjz4z 2844042 2844039 2026-11-02T22:15:21Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844042 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme|regular convex 4-polytopes]] is beyond the scope of this paper, but we may start with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the latter will build the other three 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 7pyme7v17xripxbib7hncav8b5jbgr8 2844043 2844042 2026-11-02T22:16:34Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844043 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the latter will build the other three 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} ntildbhl79ry0c3edae8jw4955zdhg9 2844044 2844043 2026-11-02T22:18:09Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844044 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element is that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the latter will build the other three 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 3i1xsqokj19xn0vo6dt3nt6998mefmr 2844045 2844044 2026-11-02T22:34:08Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844045 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the latter will build the other three 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 7zxqnbsiyosjgxx9c6q14ra37hma3jh 2844047 2844045 2026-11-02T22:50:49Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844047 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the latter will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship in the symmetries of all the 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} fx2n30xuscm5szqsa2lryqslia6fai5 2844048 2844047 2026-11-02T22:52:03Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844048 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 9vmcf6bt94u3ezpauwkt8x2a2txotli 2844049 2844048 2026-11-02T22:53:26Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844049 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme#Regular convex 4-polytopes|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 8-point 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} owgdmxdo342y4jz8gy6owtes9ci46n6 2844056 2844049 2026-11-02T23:05:36Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844056 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme#Regular convex 4-polytopes|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} kswp3pyq0n4v0n01cla7204npmfvl12 2844058 2844056 2026-11-02T23:10:12Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844058 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme#Regular convex 4-polytopes|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 7jade0ybt780rmsr96o0otfwotwq99v 2844061 2844058 2026-11-02T23:12:11Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844061 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as an <math>n</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are direct dimensional analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Objects with mass of any size (including atoms and their subatomic particles) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, such as an atom in its own proper reference frame. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences implies that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective distinct screw translations <math>QT</math>. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation <math>T</math> through 4-space. The velocity of that vector remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two mirror surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme#Regular convex 4-polytopes|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating Euclidean and spherical <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any reference frame, not only from their own proper inertial reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice."<ref name=Einstein1923/> Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassmann and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."}} Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''[[polyscheme]]'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he found all of the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|six convex regular polychora]] which can be constructed in a Euclidean space of four dimensions (the set analogous to the five [[w:Platonic solid|Platonic solids]] the ancients found in three dimensional space). Thus Schläfli was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover its astonishing regular objects. Because his work was only published posthumously in 1901, and remained almost completely unknown until Coxeter published [[w:Regular_Polytopes_(book)|Regular Polytopes]] in 1947, other researchers had more than fifty years to rediscover the regular polychora, and competing terms were coined; today [[w:Reinhold_Hoppe|Reinhold Hoppe]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme.''{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} Because of this century-long lag in the dissemination of a scientific discovery, the regular 4-polytopes appear to have played no role at all, by any name, in the twentieth century discovery and evolution of the theories of relativity and quantum mechanics.{{Efn|One could argue that the higher-dimensional polytopes have barely influenced science or culture at all thus far. The physicist John Edward Huth's comprehensive deep dive through the history of cultural and scientific concepts of physical space, from ancient flatland models of the world through general relativity and quantum mechancs, shows exactly how we got to our present standard model of the universe, although it includes no mention of higher-dimensional Euclidean space.<ref>{{Cite book|last=Huth|first=John Edward|title=A Sense of Space: A local's guide to a flat earth, the edge of the cosmos, and other curious places|year=2025|publisher=University of Chicago Press}}</ref>}} == Boundaries == <blockquote>Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three .... In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it. We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed "Dimensional Analogy."<ref>{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :We shall not cease from exploration :And the end of all our exploring :Will be to arrive where we started :And know the place for the first time. :Through the unknown, remembered gate :When the last of earth left to discover :Is that which was the beginning; :At the source of the longest river :The voice of the hidden waterfall :And the children in the apple-tree :Not known, because not looked for :But heard, half-heard, in the stillness :Between two waves of the sea. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::BEECH :Where my imaginary line :Bends square in woods, an iron spine :And pile of real rocks have been founded. :And off this corner in the wild, :Where these are driven in and piled, :One tree, by being deeply wounded, :Has been impressed as Witness Tree :And made commit to memory :My proof of being not unbounded. :Thus truth's established and borne out, :Though circumstanced with dark and doubt— :Though by a world of doubt surrounded. :::::::—''The Moodie Forester''<ref>{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} {{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's surface), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}} {{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}} {{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles]."}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}} {{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[W:Clifford parallel|Clifford parallel]] great circles spanned by a twisted [[W:Annulus (mathematics)|annulus]].]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the [[W:2-sphere|2-sphere]] will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect. In 4-polytopes various discrete sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in [[W:Hopf fibration|Hopf fiber bundles]] which visit all the vertices just once. The simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles, intersecting at 8 points defining a [[16-cell]]. Each completely orthogonal pair of circles is Clifford parallel. They cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 16-cell. Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]]|name=Clifford parallels}} {{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}} {{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}} {{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} {{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} {{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}} {{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}} {{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} {{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90^o bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60^o bend, or as a straight line with one 60^o bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}} {{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)"}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} {{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}} {{Efn|Departing from any vertex V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 64ai9mra9pyduwlppd2eu4jvd9m5dxh 0 304617 2844050 2804216 2026-11-02T22:55:22Z Dc.samizdat 2856930 /* Regular convex 4-polytopes */ 2844050 wikitext text/x-wiki {{Polyscheme}} [[File:24-cell-3CP.gif|thumb|The 24-point [[24-cell]], a 4-dimensional regular polytope, contains three 8-point [[16-cell]]s (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes. One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which in a unit-radius, unit-edge 24-cell is a {{radic|2}} edge of the red, green, or blue 16-cell.]] '''Polyscheme''' is the name given to geometric objects of any number of dimensions ([[W:Polytope|polytope]]s) by [[W:Ludwig Schläfli|Ludwig Schläfli]], the Swiss mathematician who discovered [[W:Regular polytopes (book)|all the regular polytopes]] which exist in higher dimensions of [[w:Euclidean_space|Euclidean space]] before 1853, at "a time when [[w:Arthur Cayley|Cayley]], [[w:Hermann Grassmann|Grassmann]]{{Efn|In 1844, [[w:Hermann Grassmann|Hermann Grassmann]] proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once [[W:geometry|geometry]] is put into the algebraic form he advocated now known as the [[W:Grassmannian|Grassmannian]], the number three has no privileged role as the number of spatial [[W:dimension (mathematics)|dimension]]s; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.{{Efn|[[w:Hermann Grassmann|Grassmann]] moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.}}|name=Grassmann}} and [[w:August Ferdinand Möbius|Möbius]] were the only other people who had ever conceived the possibility of [[w:Euclidean_geometry#Higher_dimensions|geometry in more than three dimensions]]."{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."}} The [[Wikiversity:Introduction|Wikiversity]] was hosted on very slow servers in those days, and other researchers also discovered the [[w:4-polytope|4-polytopes]] before Schläfli's article was published posthumously in 1901, but Schläfli is the founding author of the Polyscheme learning project.{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} [[w:H.S.M._Coxeter|H.S.M. Coxeter]] is its founding editor, whose 1948 book [[w:Regular_Polytopes|Regular Polytopes]] tells the whole story of the project. == Polyscheme learning project == The Polyscheme project is intended to be [[Polyscheme#Polyscheme project articles|a series of wiki-format articles]] on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes which pop up in context.{{Efn|[[File:Fry-lightbulb-on-forehead1.jpg|thumbnail|upright|The inventor [[W:Arthur Fry|Arthur Fry]] with a Post-it note on his forehead.]]If you hover the cursor over a footnote it will pop up in a floating box like a [[W:post-it note|post-it note]], so you can quickly get a deeper explanation of a term or a sentence you can't parse. If you click on the footnote reference, you can read a larger note like this one where it appears in the [[#Notes|Notes section]] of the article, like a mini-article within the article. It may occur in other Polyscheme project articles as well. The notes are a subject-specific hypertext of polyscheme concepts, a wiki within a wiki. From the Notes section you can see all the places where this explanatory note is cited in this article, and even go there yourself if you want to understand what depends on this concept. Many of the explanatory notes contain footnote references themselves, to other explanatory notes. You can go as far down this rabbit hole as you need to go for comprehension, but beware of getting lost underground in a [[W:Twisty little maze of passages|twisty little maze of passages]]! At least with footnotes there is no danger of leaving the article altogether, and never coming back to finish what you started.|name=explanatory notes}} Some of what is in [[#Polyscheme project articles|these companion articles]] is the result of original research that may not have been peer reviewed yet, and so has the status of opinion as of this date of publication, and some of it is commentary, not essential fact. The commentary and recent research is precisely the difference between an expanded Wikiversity learning project article and the corresponding Wikipedia encyclopedia article; you can compare them to detect it, or just read the encyclopedia instead if you don't trust it. Most project articles are an annotated and '''expanded version''' of the Wikipedia article, and they replace it for learning purposes. Some project articles, however, do not reproduce the Wikipedia article. Their banner indicates that they are only a '''commentary''' on it, and participants are directed to the Wikipedia article by a "see also" link below the banner. == Active research == Polyschemes have been a subject of active and ongoing research since their discovery before 1853 by the Swiss researcher [[W:Ludwig Schläfli|Ludwig Schläfli]], who was the first person to extend Euclidean geometry by '''dimensional analogy''' beyond three dimensions. But for the first 50 years of the subject's history Schläfli's paper was unpublished, entirely inaccessible to other researchers. Even after its posthumous publication, Schläfli's paper remained obscure for another 50 years, in part because the mathematics it contained was only accessible to a few mathematicians who could read that specialized language. During this period other researchers also discovered higher-dimensional Euclidean space. H.S.M. Coxeter finally made the subject widely accessible in his 1948 book [[W:Regular Polytopes (book)|Regular Polytopes]], which synthesized all the research that had been done since Schläfli and added Coxeter's discoveries, including his invention of the [[w:Coxeter_group|theory of reflecting symmetry groups]], the [[w:Group_theory|group theory mathematics]] that underlies geometry. Since then, Coxeter's book has been the definitive work on [[W:Euclidean geometry|Euclidean geometry]] in ''n''-dimensional space, and every polyscheme researcher has been able to begin with it instead of reinventing the wheel, and contribute new observations to it. == Polyscheme project articles == The following Wikiversity articles comprise the Polyscheme collection of Wikipedia companion articles at present. Please add your research contributions to them, and consider adding to this list of Polyscheme articles, if you have a research contribution to make to another relevant Wikipedia article, that would not be appropriate in the encyclopedia, or you have written an original research article on the subject. === Regular convex 4-polytopes === *5-point [[5-cell]] 4-simplex *8-point [[16-cell]] 4-orthoplex *24-point [[24-cell]] *120-point [[600-cell]] *600-point [[120-cell]] *[[Template:Regular convex 4-polytopes]] === Uniform 3-polytopes === *[[Kinematics of the cuboctahedron]] === Other conceptual objects === *[[Schläfli double six]] ===See also=== *[[w:Tesseract|8-cell (tesseract)]] *[[11-cell]] === Sequence of regular 4-polytopes === {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} [[Category:Geometry]] [[Category:Polyscheme]] lz69ozeimagibv09gvk4qjllizm44r6 2844052 2844050 2026-11-02T22:59:29Z Dc.samizdat 2856930 /* Regular convex 4-polytopes */ 2844052 wikitext text/x-wiki {{Polyscheme}} [[File:24-cell-3CP.gif|thumb|The 24-point [[24-cell]], a 4-dimensional regular polytope, contains three 8-point [[16-cell]]s (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes. One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which in a unit-radius, unit-edge 24-cell is a {{radic|2}} edge of the red, green, or blue 16-cell.]] '''Polyscheme''' is the name given to geometric objects of any number of dimensions ([[W:Polytope|polytope]]s) by [[W:Ludwig Schläfli|Ludwig Schläfli]], the Swiss mathematician who discovered [[W:Regular polytopes (book)|all the regular polytopes]] which exist in higher dimensions of [[w:Euclidean_space|Euclidean space]] before 1853, at "a time when [[w:Arthur Cayley|Cayley]], [[w:Hermann Grassmann|Grassmann]]{{Efn|In 1844, [[w:Hermann Grassmann|Hermann Grassmann]] proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once [[W:geometry|geometry]] is put into the algebraic form he advocated now known as the [[W:Grassmannian|Grassmannian]], the number three has no privileged role as the number of spatial [[W:dimension (mathematics)|dimension]]s; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.{{Efn|[[w:Hermann Grassmann|Grassmann]] moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.}}|name=Grassmann}} and [[w:August Ferdinand Möbius|Möbius]] were the only other people who had ever conceived the possibility of [[w:Euclidean_geometry#Higher_dimensions|geometry in more than three dimensions]]."{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."}} The [[Wikiversity:Introduction|Wikiversity]] was hosted on very slow servers in those days, and other researchers also discovered the [[w:4-polytope|4-polytopes]] before Schläfli's article was published posthumously in 1901, but Schläfli is the founding author of the Polyscheme learning project.{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} [[w:H.S.M._Coxeter|H.S.M. Coxeter]] is its founding editor, whose 1948 book [[w:Regular_Polytopes|Regular Polytopes]] tells the whole story of the project. == Polyscheme learning project == The Polyscheme project is intended to be [[Polyscheme#Polyscheme project articles|a series of wiki-format articles]] on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes which pop up in context.{{Efn|[[File:Fry-lightbulb-on-forehead1.jpg|thumbnail|upright|The inventor [[W:Arthur Fry|Arthur Fry]] with a Post-it note on his forehead.]]If you hover the cursor over a footnote it will pop up in a floating box like a [[W:post-it note|post-it note]], so you can quickly get a deeper explanation of a term or a sentence you can't parse. If you click on the footnote reference, you can read a larger note like this one where it appears in the [[#Notes|Notes section]] of the article, like a mini-article within the article. It may occur in other Polyscheme project articles as well. The notes are a subject-specific hypertext of polyscheme concepts, a wiki within a wiki. From the Notes section you can see all the places where this explanatory note is cited in this article, and even go there yourself if you want to understand what depends on this concept. Many of the explanatory notes contain footnote references themselves, to other explanatory notes. You can go as far down this rabbit hole as you need to go for comprehension, but beware of getting lost underground in a [[W:Twisty little maze of passages|twisty little maze of passages]]! At least with footnotes there is no danger of leaving the article altogether, and never coming back to finish what you started.|name=explanatory notes}} Some of what is in [[#Polyscheme project articles|these companion articles]] is the result of original research that may not have been peer reviewed yet, and so has the status of opinion as of this date of publication, and some of it is commentary, not essential fact. The commentary and recent research is precisely the difference between an expanded Wikiversity learning project article and the corresponding Wikipedia encyclopedia article; you can compare them to detect it, or just read the encyclopedia instead if you don't trust it. Most project articles are an annotated and '''expanded version''' of the Wikipedia article, and they replace it for learning purposes. Some project articles, however, do not reproduce the Wikipedia article. Their banner indicates that they are only a '''commentary''' on it, and participants are directed to the Wikipedia article by a "see also" link below the banner. == Active research == Polyschemes have been a subject of active and ongoing research since their discovery before 1853 by the Swiss researcher [[W:Ludwig Schläfli|Ludwig Schläfli]], who was the first person to extend Euclidean geometry by '''dimensional analogy''' beyond three dimensions. But for the first 50 years of the subject's history Schläfli's paper was unpublished, entirely inaccessible to other researchers. Even after its posthumous publication, Schläfli's paper remained obscure for another 50 years, in part because the mathematics it contained was only accessible to a few mathematicians who could read that specialized language. During this period other researchers also discovered higher-dimensional Euclidean space. H.S.M. Coxeter finally made the subject widely accessible in his 1948 book [[W:Regular Polytopes (book)|Regular Polytopes]], which synthesized all the research that had been done since Schläfli and added Coxeter's discoveries, including his invention of the [[w:Coxeter_group|theory of reflecting symmetry groups]], the [[w:Group_theory|group theory mathematics]] that underlies geometry. Since then, Coxeter's book has been the definitive work on [[W:Euclidean geometry|Euclidean geometry]] in ''n''-dimensional space, and every polyscheme researcher has been able to begin with it instead of reinventing the wheel, and contribute new observations to it. == Polyscheme project articles == The following Wikiversity articles comprise the Polyscheme collection of Wikipedia companion articles at present. Please add your research contributions to them, and consider adding to this list of Polyscheme articles, if you have a research contribution to make to another relevant Wikipedia article, that would not be appropriate in the encyclopedia, or you have written an original research article on the subject. === Regular convex 4-polytopes === *5-point [[5-cell]] 4-simplex *8-point [[16-cell]] 4-orthoplex *16-point [[W:8-cell|8-cell]] 4-hypercube (no Wikiversity article, see this Wikipedia article) *24-point [[24-cell]] *120-point [[600-cell]] *600-point [[120-cell]] *[[Template:Regular convex 4-polytopes]] === Uniform 3-polytopes === *[[Kinematics of the cuboctahedron]] === Other conceptual objects === *[[Schläfli double six]] ===See also=== *[[w:Tesseract|8-cell (tesseract)]] *[[11-cell]] === Sequence of regular 4-polytopes === {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} [[Category:Geometry]] [[Category:Polyscheme]] r8jjmqjvrohp0udpxf73omku76zjd96 2844053 2844052 2026-11-02T23:00:25Z Dc.samizdat 2856930 /* Regular convex 4-polytopes */ 2844053 wikitext text/x-wiki {{Polyscheme}} [[File:24-cell-3CP.gif|thumb|The 24-point [[24-cell]], a 4-dimensional regular polytope, contains three 8-point [[16-cell]]s (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes, just as a 6-point [[w:Octahedron|octahedron]] is a coordinate system basis frame of three perpendicular (x,y,z) axes. One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which in a unit-radius, unit-edge 24-cell is a {{radic|2}} edge of the red, green, or blue 16-cell.]] '''Polyscheme''' is the name given to geometric objects of any number of dimensions ([[W:Polytope|polytope]]s) by [[W:Ludwig Schläfli|Ludwig Schläfli]], the Swiss mathematician who discovered [[W:Regular polytopes (book)|all the regular polytopes]] which exist in higher dimensions of [[w:Euclidean_space|Euclidean space]] before 1853, at "a time when [[w:Arthur Cayley|Cayley]], [[w:Hermann Grassmann|Grassmann]]{{Efn|In 1844, [[w:Hermann Grassmann|Hermann Grassmann]] proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once [[W:geometry|geometry]] is put into the algebraic form he advocated now known as the [[W:Grassmannian|Grassmannian]], the number three has no privileged role as the number of spatial [[W:dimension (mathematics)|dimension]]s; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.{{Efn|[[w:Hermann Grassmann|Grassmann]] moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.}}|name=Grassmann}} and [[w:August Ferdinand Möbius|Möbius]] were the only other people who had ever conceived the possibility of [[w:Euclidean_geometry#Higher_dimensions|geometry in more than three dimensions]]."{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."}} The [[Wikiversity:Introduction|Wikiversity]] was hosted on very slow servers in those days, and other researchers also discovered the [[w:4-polytope|4-polytopes]] before Schläfli's article was published posthumously in 1901, but Schläfli is the founding author of the Polyscheme learning project.{{Efn|[[w:Reinhold_Hoppe|Reinhold Hoppe]]'s German word ''polytop'' was introduced into English by [[W:Alicia Boole Stott|Alicia Boole Stott]], who like Hoppe and [[W:Thorold Gosset|Thorold Gosset]] rediscovered Schlafli's six regular convex 4-polytopes, with no knowledge of their prior discovery. Today Schläfli's original ''polyschem'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}} [[w:H.S.M._Coxeter|H.S.M. Coxeter]] is its founding editor, whose 1948 book [[w:Regular_Polytopes|Regular Polytopes]] tells the whole story of the project. == Polyscheme learning project == The Polyscheme project is intended to be [[Polyscheme#Polyscheme project articles|a series of wiki-format articles]] on the [[W:Regular polytope|regular polytope]]s, the [[W:Four-dimensional space|fourth spatial dimension]], and the general dimensional analogy of [[W:Euclidean space|Euclidean]] and [[W:n-sphere|spherical space]]s of any number of [[W:Dimension|dimension]]s. This series of articles expands the corresponding Wikipedia encyclopedia articles, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes which pop up in context.{{Efn|[[File:Fry-lightbulb-on-forehead1.jpg|thumbnail|upright|The inventor [[W:Arthur Fry|Arthur Fry]] with a Post-it note on his forehead.]]If you hover the cursor over a footnote it will pop up in a floating box like a [[W:post-it note|post-it note]], so you can quickly get a deeper explanation of a term or a sentence you can't parse. If you click on the footnote reference, you can read a larger note like this one where it appears in the [[#Notes|Notes section]] of the article, like a mini-article within the article. It may occur in other Polyscheme project articles as well. The notes are a subject-specific hypertext of polyscheme concepts, a wiki within a wiki. From the Notes section you can see all the places where this explanatory note is cited in this article, and even go there yourself if you want to understand what depends on this concept. Many of the explanatory notes contain footnote references themselves, to other explanatory notes. You can go as far down this rabbit hole as you need to go for comprehension, but beware of getting lost underground in a [[W:Twisty little maze of passages|twisty little maze of passages]]! At least with footnotes there is no danger of leaving the article altogether, and never coming back to finish what you started.|name=explanatory notes}} Some of what is in [[#Polyscheme project articles|these companion articles]] is the result of original research that may not have been peer reviewed yet, and so has the status of opinion as of this date of publication, and some of it is commentary, not essential fact. The commentary and recent research is precisely the difference between an expanded Wikiversity learning project article and the corresponding Wikipedia encyclopedia article; you can compare them to detect it, or just read the encyclopedia instead if you don't trust it. Most project articles are an annotated and '''expanded version''' of the Wikipedia article, and they replace it for learning purposes. Some project articles, however, do not reproduce the Wikipedia article. Their banner indicates that they are only a '''commentary''' on it, and participants are directed to the Wikipedia article by a "see also" link below the banner. == Active research == Polyschemes have been a subject of active and ongoing research since their discovery before 1853 by the Swiss researcher [[W:Ludwig Schläfli|Ludwig Schläfli]], who was the first person to extend Euclidean geometry by '''dimensional analogy''' beyond three dimensions. But for the first 50 years of the subject's history Schläfli's paper was unpublished, entirely inaccessible to other researchers. Even after its posthumous publication, Schläfli's paper remained obscure for another 50 years, in part because the mathematics it contained was only accessible to a few mathematicians who could read that specialized language. During this period other researchers also discovered higher-dimensional Euclidean space. H.S.M. Coxeter finally made the subject widely accessible in his 1948 book [[W:Regular Polytopes (book)|Regular Polytopes]], which synthesized all the research that had been done since Schläfli and added Coxeter's discoveries, including his invention of the [[w:Coxeter_group|theory of reflecting symmetry groups]], the [[w:Group_theory|group theory mathematics]] that underlies geometry. Since then, Coxeter's book has been the definitive work on [[W:Euclidean geometry|Euclidean geometry]] in ''n''-dimensional space, and every polyscheme researcher has been able to begin with it instead of reinventing the wheel, and contribute new observations to it. == Polyscheme project articles == The following Wikiversity articles comprise the Polyscheme collection of Wikipedia companion articles at present. Please add your research contributions to them, and consider adding to this list of Polyscheme articles, if you have a research contribution to make to another relevant Wikipedia article, that would not be appropriate in the encyclopedia, or you have written an original research article on the subject. === Regular convex 4-polytopes === *5-point [[5-cell]] 4-simplex *8-point [[16-cell]] 4-orthoplex *16-point [[W:8-cell|8-cell]] 4-hypercube (no Wikiversity article yet, see this Wikipedia article) *24-point [[24-cell]] *120-point [[600-cell]] *600-point [[120-cell]] *[[Template:Regular convex 4-polytopes]] === Uniform 3-polytopes === *[[Kinematics of the cuboctahedron]] === Other conceptual objects === *[[Schläfli double six]] ===See also=== *[[w:Tesseract|8-cell (tesseract)]] *[[11-cell]] === Sequence of regular 4-polytopes === {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} [[Category:Geometry]] [[Category:Polyscheme]] 74tvnsz4l3ubxc3swnfdeuqtczog2lr 0 305491 2844002 2843878 2026-11-02T18:23:22Z ~2026-56983-83 3113232 2844002 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1876 (9) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1877-1911 (35) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (14) | |- |Silent |1925-1945 (21) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1934 (9) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1945 (22) |- |Baby boom |1946-1964 (19) |1946-1964 (19) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1946-1965 (20) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1965-1984 (20) |- |Millennial |1980-1994 (15) |1981-1996 (16) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1985-2005 (21) |- |Generation Z |1995-2009 (15) |1997-2012 (16) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2004 (8) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2013-? |2006-2029 (24) |TBD |2006-mid 2020s (19-21*) |- |Generation Beta |2025-2039 (15) |TBD |2030-early 2050s (20-25*) |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" !Birth year range !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary |- |1433-1460 | rowspan="20" |N/A | rowspan="21" |N/A |Arthurian | colspan="2" |Arthurian |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1882 |Missionary | colspan="2" |Missionary |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1924 |G.I. | colspan="2" |G.I |- |1925-1927 | rowspan="3" |Builder | rowspan="2" |Silent |G.I !Silent |- |1928-1942 | rowspan="2" |Silent | colspan="2" |Silent |- |1943-1945 | rowspan="2" |Baby Boom !Silent |Baby Boom |- |1946-1960 | rowspan="2" |Baby Boom | rowspan="2" |Baby Boom | colspan="2" |Baby Boom |- |1961-1964 | rowspan="4" |Generation X !Baby Boom |Generation X |- |1965-1979 |Generation X | rowspan="2" |Generation X | colspan="2" |Generation X |- |1980 | rowspan="3" |Generation Y !Generation X |Millennial |- |1981 | rowspan="3" |Millennial |Generation X !Millennial |- |1982-1994 | rowspan="3" |Millennial | colspan="2" |Millennial |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |- |2006-2009 | rowspan="4" |Homeland !Generation Z |Generation Alpha |- |2010-2012 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha |- |2013-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD |- |2030-2039 |TBD |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1928-1942 (15) |1925-1945 (21) |1925-1945 (21) |- |Baby Boom |1946-1960 (15) |1946-1964 (19) |1943-1964 (22) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |- |Millennial |1982-1994 (13) |1981-1996 (16) |1980-2005 (26) |- |Generation Z |N/A |1997-2009 (13) |1995-2012 (18) |- |Generation Alpha |2013-2024 (12) |2010-? |2006-2029 (24) |- |Generation Beta |TBD |TBD |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Gen Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf sx2aphbzhy0gky9856borz5t7ot6fw4 2844163 2844002 2026-11-03T10:43:17Z ~2026-56983-83 3113232 /* */ 2844163 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1876 (9) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1877-1911 (35) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (14) | |- |Silent |1925-1945 (21) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1934 (9) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1945 (22) |- |Baby Boom |1946-1964 (19) |1946-1964 (19) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1946-1965 (20) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1965-1984 (20) |- |Millennial |1980-1994 (15) |1981-1996 (16) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1985-2005 (21) |- |Generation Z |1995-2009 (15) |1997-2012 (16) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2004 (8) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2013-? |2006-2029 (24) |TBD |2006-mid 2020s (19-21*) |- |Generation Beta |2025-2039 (15) |TBD |2030-early 2050s (20-25*) |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" !Birth year range !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary |- |1433-1460 | rowspan="20" |N/A | rowspan="21" |N/A |Arthurian | colspan="2" |Arthurian |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1882 |Missionary | colspan="2" |Missionary |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1924 |G.I. | colspan="2" |G.I |- |1925-1927 | rowspan="3" |Builder | rowspan="2" |Silent |G.I !Silent |- |1928-1942 | rowspan="2" |Silent | colspan="2" |Silent |- |1943-1945 | rowspan="2" |Baby Boom !Silent |Baby Boom |- |1946-1960 | rowspan="2" |Baby Boom | rowspan="2" |Baby Boom | colspan="2" |Baby Boom |- |1961-1964 | rowspan="4" |Generation X !Baby Boom |Generation X |- |1965-1979 |Generation X | rowspan="2" |Generation X | colspan="2" |Generation X |- |1980 | rowspan="3" |Generation Y !Generation X |Millennial |- |1981 | rowspan="3" |Millennial |Generation X !Millennial |- |1982-1994 | rowspan="3" |Millennial | colspan="2" |Millennial |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |- |2006-2009 | rowspan="4" |Homeland !Generation Z |Generation Alpha |- |2010-2012 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha |- |2013-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD |- |2030-2039 |TBD |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1928-1942 (15) |1925-1945 (21) |1925-1945 (21) |- |Baby Boom |1946-1960 (15) |1946-1964 (19) |1943-1964 (22) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |- |Millennial |1982-1994 (13) |1981-1996 (16) |1980-2005 (26) |- |Generation Z |N/A |1997-2009 (13) |1995-2012 (18) |- |Generation Alpha |2013-2024 (12) |2010-? |2006-2029 (24) |- |Generation Beta |TBD |TBD |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Gen Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf e11yukw7dm2cxrcxs8cxp1phub12id8 1 307245 2844145 2841657 2026-11-03T09:05:03Z Jtneill 10242 2844145 wikitext text/x-wiki == Helpful peer reviewed APA 7 article for Hygiene Motivation if you wish to add or remove!! == The concept of hygiene motivation is examined in the study by Curtis, V., & Cairncross, S. (2003), which emphasises that people's hygiene behaviours are primarily driven by the desire to avoid disgust rather than by health education alone. The authors contend that disgust is a potent motivator that can be used in public health campaigns to promote better hygiene practices, particularly in prevention of disease transmission. The study underscores the significance of comprehending emotional drivers like disgust in designing more successful health interventions. Curtis, V., & Cairncross, S. (2003). Effect of washing hands with soap on diarrhoea risk in the community: a systematic review. ''The Lancet Infectious Diseases'', ''3''(5), 275–281. <nowiki>https://doi.org/10.1016/S1473-3099(03)00606-6</nowiki> [[User:U3214564|U3214564]] ([[User talk:U3214564|discuss]] • [[Special:Contributions/U3214564|contribs]]) 08:19, 15 August 2024 (UTC) == Feedback and Suggestions == Hey Jack, I really like your book chapter so far! I found this article from Miller in 2020 which covers motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak. I think this would be an interesting topic to explore and touch on within your book chapter. This article focusses primarily on the role of human behaviour in the COVID-19 outbreak and the speed of the spread across the world, and how human hygiene motivation was associated with the progression. Here is the reference to the article: Gibson Miller, J. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British Journal of Health Psychology'', ''25''(4), 856–864. <nowiki>https://doi.org/10.1111/bjhp.12426</nowiki> Good luck! :) - Alyssia [[User:Alyssia Myers|Alyssia Myers]] ([[User talk:Alyssia Myers|discuss]] • [[Special:Contributions/Alyssia Myers|contribs]]) 11:15, 15 August 2024 (UTC) Hello Jack, it looks like you are hard at work on your page, I've adding another figure to your page, hopefully it'll help you flesh out your article !![https://commons.wikimedia.org/wiki/File:OCD_handwash.jpg this is the image], you could use it to talk about personal hygiene, such as washing one's hands !!! Good luck [[User:Gabriel Geld|Gabriel Geld]] ([[User talk:Gabriel Geld|discuss]] • [[Special:Contributions/Gabriel Geld|contribs]]) 08:43, 16 August 2024 (UTC) <!-- Official topic development feedback --> {{METF/2024 |1= <!-- Title --> # The title and sub-title are correctly worded; casing has been corrected # I've abbreviated the sub-title (same meaning) |2= <!-- Headings --> # Well developed 2-level heading structure. Meaningful headings clearly relate directly to the core topic. # Consider abbreviating the two introductory sections (2 and 3) into a single section; it is OK to retain but keep these sections relatively brief, with embedded links to further information so that the chapter concentrates on psychological science # Remove trailing colons <!-- Alignment with focus questions --> # Good alignment between focus questions and heading structure |3= <!-- Overview--> <!-- Scenario --> # A scenario or case study is presented in a feature box at the start of this section # Add an image to the scenario or case study to help attract reader interest <!-- Description --> # Add a brief, evocative description of the problem/topic <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings # Use sentence casing (Hygiene -> hygiene) # Move quiz into a subsequent section |4= <!-- Key points--> # Partial development of key points for some sections, with limited use of relevant citations <!-- Theory and research --> # Strive for an integrated balance of theory and research, with practical examples # It is unclear whether the best available psychological theory and research has been consulted in the preparation of this plan <!-- Conclusion --> # Conclusion (the most important section): ## Hasn't been developed ## What might the take-home, practical messages be? (What are the answer(s) to the question(s) in the sub-title and/or focus questions?) |5= <!-- Figure --> # A relevant figure is presented, captioned, and cited # The figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Caption citations should refer to Figure rather than figure (APA style) |6= <!-- Learning feature --> <!-- Interwiki links ---> # One use of in-text [[m:Help:Interwiki linking|interwiki links]] for the first mention of key terms to relevant Wikipedia articles and/or to other relevant book chapters <!-- Examples/case studies --> # Promising use of example(s)/case study(ies) <!-- Other --> # Consider including more examples/case studies, quiz question(s), table(s) etc. |7= <!-- References --> <!-- Overall --> # Insufficient <!-- Suggestions --> # Move non-academic / non-peer reviewed sources to External links # Are there any systematic reviews about this topic? |8= <!-- Resources --> # See also ## Not developed # External links ## Not developed |9= <!-- User page --> # Used effectively <!-- Description about self --> # Brief description about self provided – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # Link provided to book chapter |10= <!-- Social contribution --> # At least three different types of contributions with direct link(s) to evidence # The links to evidence can be improved (i.e., be more direct/accurate). For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:53, 18 August 2024 (UTC) == social contribution == hey, I like the case study you have used. linking it to how hygiene decline directly impacts him emotionally and mentally. You could expand on the motivation factors leading to this decline. you could link it to psychological theories and discuss motivational theories like Maslow’s hierarchy of needs or self-determination theory in relation to hygiene maintenance to add to the depth of your chapter. you could also expand on methods to increase hygiene motivation with actionable steps. Great work so far and good luck :) [[User:U3236683|U3236683]] ([[User talk:U3236683|discuss]] • [[Special:Contributions/U3236683|contribs]]) 07:00, 3 October 2024 (UTC) == General tips == Hello, Here are some suggestions to enhance your chapter: First, consider simplifying John’s scenario to make it more relatable. Clearly outline how his hygiene standards declined due to increased stress. This will help readers connect with his situation better. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1.r. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1. [[User:U3219927|U3219927]] ([[User talk:U3219927|discuss]] • [[Special:Contributions/U3219927|contribs]]) 11:35, 6 October 2024 (UTC) == Helpful references == Hi there, I have a suggestion for some references you may like to look at for your topic! '''Motivation Hygiene Theory:''' Brenner, V. C., Carmack, C. W., & Weinstein, M. G. (1971). An empirical test of the motivation-hygiene theory. ''Journal of Accounting Research'', 359-366. '''Application to real world & COVID:''' Gibson Miller, J., Hartman, T. K., Levita, L., Martinez, A. P., Mason, L., McBride, O., ... & Bentall, R. P. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British journal of health psychology'', ''25''(4), 856-864. [[User:Sienna33309|Sienna33309]] ([[User talk:Sienna33309|discuss]] • [[Special:Contributions/Sienna33309|contribs]]) 02:53, 20 August 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] <!-- Alignment with focus questions --> # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # The COVID section could be a subheading |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings # Use open- rather then close-ended focus questions |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad), but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title <!-- Theory and research --> # Promising balance of theory and research <!-- Conclusion --> # Conclusion is well underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) <!-- Tables --> # Excellent use of additional images # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # Only cite sources that you have consulted <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## italicisation ## use dois where available instead of other links ## include hyperlinked dois ## use the simplest, direct link for sources without dois |8= <!-- Resources --> <!-- See also --> # See also ## Basic ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Excellent |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:22, 31 August 2026 (UTC) == Feedback == Hi! I really like your chapter so far and it's quite clear and easy to understand. The only thing I was wondering whilst reading it was whether the sub heading for the health belief model should be closer to the top since the chapter is based off that and also cause it was mentioned quite early on in the chapter. Just a thought, it's very good so far! [[User:U3280499|U3280499]] ([[User talk:U3280499|discuss]] • [[Special:Contributions/U3280499|contribs]]) 08:49, 8 October 2026 (UTC) [[User:U3280499|U3280499]] ([[User talk:U3280499|discuss]] • [[Special:Contributions/U3280499|contribs]]) 08:50, 8 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # The main area(s) for potential improvement: #* more integrative review of the best psychological research about the topic #* quality of written expression #* better citation of sources using APA style #* integrate more strongly with other chapters by using embedded links to related topics <!-- Overall - Edit summaries --> #* Insufficient use [[w:Help:Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FHygiene_motivation&diff=2841655&oldid=2841264 these copyedits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied <!-- Theory – Builds on --> # Builds effectively on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Very good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags), incluidng the year of publication. <!-- Theory – Examples --> # Very good use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Very good review of relevant research # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # More detail about key studies would be ideal # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). <!-- Research – Critical thinking --> # Basic [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Very good integration between the most relevant theory(ies) and the best research # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence |6= <!-- Conclusion comments... --> # Very good summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is very good ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) <!-- Written expression – Grammar --> # Grammar ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – APA style --> # APA style ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – Figures --> ## Figures ### Well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Tables --> ## Tables ### Well captioned ### Use APA style captions (capitalisation) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] because claims are made without citations and the citation year is often missing ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources <!-- Written expression – References --> ## References use excellent APA style: <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Very good use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Good use of figure(s) # Excellent use of table(s) # Basic use of feature box(es) # Good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Very good use of quiz(zes) and/or reflection question(s) emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section ## Add Wikipedia links <!-- Learning features – External links --> # Basic use of the [[{{PAGENAME}}#External links|External links]] section ## Target an international audience ## Add more links |9= <!-- Social contribution comments... --> # ~5 logged, useful, mostly minor to moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] }} 8qdfe0pt9zwarlqpnzdacgk2b9d4gk3 2844146 2844145 2026-11-03T09:06:56Z Jtneill 10242 Fix display 2844146 wikitext text/x-wiki == Helpful peer reviewed APA 7 article for Hygiene Motivation if you wish to add or remove!! == The concept of hygiene motivation is examined in the study by Curtis, V., & Cairncross, S. (2003), which emphasises that people's hygiene behaviours are primarily driven by the desire to avoid disgust rather than by health education alone. The authors contend that disgust is a potent motivator that can be used in public health campaigns to promote better hygiene practices, particularly in prevention of disease transmission. The study underscores the significance of comprehending emotional drivers like disgust in designing more successful health interventions. Curtis, V., & Cairncross, S. (2003). Effect of washing hands with soap on diarrhoea risk in the community: a systematic review. ''The Lancet Infectious Diseases'', ''3''(5), 275–281. <nowiki>https://doi.org/10.1016/S1473-3099(03)00606-6</nowiki> [[User:U3214564|U3214564]] ([[User talk:U3214564|discuss]] • [[Special:Contributions/U3214564|contribs]]) 08:19, 15 August 2024 (UTC) == Feedback and Suggestions == Hey Jack, I really like your book chapter so far! I found this article from Miller in 2020 which covers motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak. I think this would be an interesting topic to explore and touch on within your book chapter. This article focusses primarily on the role of human behaviour in the COVID-19 outbreak and the speed of the spread across the world, and how human hygiene motivation was associated with the progression. Here is the reference to the article: Gibson Miller, J. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British Journal of Health Psychology'', ''25''(4), 856–864. <nowiki>https://doi.org/10.1111/bjhp.12426</nowiki> Good luck! :) - Alyssia [[User:Alyssia Myers|Alyssia Myers]] ([[User talk:Alyssia Myers|discuss]] • [[Special:Contributions/Alyssia Myers|contribs]]) 11:15, 15 August 2024 (UTC) Hello Jack, it looks like you are hard at work on your page, I've adding another figure to your page, hopefully it'll help you flesh out your article !![https://commons.wikimedia.org/wiki/File:OCD_handwash.jpg this is the image], you could use it to talk about personal hygiene, such as washing one's hands !!! Good luck [[User:Gabriel Geld|Gabriel Geld]] ([[User talk:Gabriel Geld|discuss]] • [[Special:Contributions/Gabriel Geld|contribs]]) 08:43, 16 August 2024 (UTC) <!-- Official topic development feedback --> {{METF/2024 |1= <!-- Title --> # The title and sub-title are correctly worded; casing has been corrected # I've abbreviated the sub-title (same meaning) |2= <!-- Headings --> # Well developed 2-level heading structure. Meaningful headings clearly relate directly to the core topic. # Consider abbreviating the two introductory sections (2 and 3) into a single section; it is OK to retain but keep these sections relatively brief, with embedded links to further information so that the chapter concentrates on psychological science # Remove trailing colons <!-- Alignment with focus questions --> # Good alignment between focus questions and heading structure |3= <!-- Overview--> <!-- Scenario --> # A scenario or case study is presented in a feature box at the start of this section # Add an image to the scenario or case study to help attract reader interest <!-- Description --> # Add a brief, evocative description of the problem/topic <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings # Use sentence casing (Hygiene -> hygiene) # Move quiz into a subsequent section |4= <!-- Key points--> # Partial development of key points for some sections, with limited use of relevant citations <!-- Theory and research --> # Strive for an integrated balance of theory and research, with practical examples # It is unclear whether the best available psychological theory and research has been consulted in the preparation of this plan <!-- Conclusion --> # Conclusion (the most important section): ## Hasn't been developed ## What might the take-home, practical messages be? (What are the answer(s) to the question(s) in the sub-title and/or focus questions?) |5= <!-- Figure --> # A relevant figure is presented, captioned, and cited # The figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Caption citations should refer to Figure rather than figure (APA style) |6= <!-- Learning feature --> <!-- Interwiki links ---> # One use of in-text [[m:Help:Interwiki linking|interwiki links]] for the first mention of key terms to relevant Wikipedia articles and/or to other relevant book chapters <!-- Examples/case studies --> # Promising use of example(s)/case study(ies) <!-- Other --> # Consider including more examples/case studies, quiz question(s), table(s) etc. |7= <!-- References --> <!-- Overall --> # Insufficient <!-- Suggestions --> # Move non-academic / non-peer reviewed sources to External links # Are there any systematic reviews about this topic? |8= <!-- Resources --> # See also ## Not developed # External links ## Not developed |9= <!-- User page --> # Used effectively <!-- Description about self --> # Brief description about self provided – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # Link provided to book chapter |10= <!-- Social contribution --> # At least three different types of contributions with direct link(s) to evidence # The links to evidence can be improved (i.e., be more direct/accurate). For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:53, 18 August 2024 (UTC) == social contribution == hey, I like the case study you have used. linking it to how hygiene decline directly impacts him emotionally and mentally. You could expand on the motivation factors leading to this decline. you could link it to psychological theories and discuss motivational theories like Maslow’s hierarchy of needs or self-determination theory in relation to hygiene maintenance to add to the depth of your chapter. you could also expand on methods to increase hygiene motivation with actionable steps. Great work so far and good luck :) [[User:U3236683|U3236683]] ([[User talk:U3236683|discuss]] • [[Special:Contributions/U3236683|contribs]]) 07:00, 3 October 2024 (UTC) == General tips == Hello, Here are some suggestions to enhance your chapter: First, consider simplifying John’s scenario to make it more relatable. Clearly outline how his hygiene standards declined due to increased stress. This will help readers connect with his situation better. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1.r. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1. [[User:U3219927|U3219927]] ([[User talk:U3219927|discuss]] • [[Special:Contributions/U3219927|contribs]]) 11:35, 6 October 2024 (UTC) == Helpful references == Hi there, I have a suggestion for some references you may like to look at for your topic! '''Motivation Hygiene Theory:''' Brenner, V. C., Carmack, C. W., & Weinstein, M. G. (1971). An empirical test of the motivation-hygiene theory. ''Journal of Accounting Research'', 359-366. '''Application to real world & COVID:''' Gibson Miller, J., Hartman, T. K., Levita, L., Martinez, A. P., Mason, L., McBride, O., ... & Bentall, R. P. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British journal of health psychology'', ''25''(4), 856-864. [[User:Sienna33309|Sienna33309]] ([[User talk:Sienna33309|discuss]] • [[Special:Contributions/Sienna33309|contribs]]) 02:53, 20 August 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] <!-- Alignment with focus questions --> # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # The COVID section could be a subheading |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings # Use open- rather then close-ended focus questions |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad), but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title <!-- Theory and research --> # Promising balance of theory and research <!-- Conclusion --> # Conclusion is well underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) <!-- Tables --> # Excellent use of additional images # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # Only cite sources that you have consulted <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## italicisation ## use dois where available instead of other links ## include hyperlinked dois ## use the simplest, direct link for sources without dois |8= <!-- Resources --> <!-- See also --> # See also ## Basic ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Excellent |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:22, 31 August 2026 (UTC) == Feedback == Hi! I really like your chapter so far and it's quite clear and easy to understand. The only thing I was wondering whilst reading it was whether the sub heading for the health belief model should be closer to the top since the chapter is based off that and also cause it was mentioned quite early on in the chapter. Just a thought, it's very good so far! [[User:U3280499|U3280499]] ([[User talk:U3280499|discuss]] • [[Special:Contributions/U3280499|contribs]]) 08:49, 8 October 2026 (UTC) [[User:U3280499|U3280499]] ([[User talk:U3280499|discuss]] • [[Special:Contributions/U3280499|contribs]]) 08:50, 8 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # The main area(s) for potential improvement: #* more integrative review of the best psychological research about the topic #* quality of written expression #* better citation of sources using APA style #* integrate more strongly with other chapters by using embedded links to related topics <!-- Overall - Edit summaries --> #* Insufficient use [[w:Help:Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FHygiene_motivation&diff=2841655&oldid=2841264 these copyedits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied <!-- Theory – Builds on --> # Builds effectively on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Very good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags), incluidng the year of publication. <!-- Theory – Examples --> # Very good use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Very good review of relevant research # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # More detail about key studies would be ideal # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). <!-- Research – Critical thinking --> # Basic [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Very good integration between the most relevant theory(ies) and the best research # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence |6= <!-- Conclusion comments... --> # Very good summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is very good ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) <!-- Written expression – Grammar --> # Grammar ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – APA style --> # APA style ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – Figures --> ## Figures ### Well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Tables --> ## Tables ### Well captioned ### Use APA style captions (capitalisation) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] because claims are made without citations and the citation year is often missing ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources <!-- Written expression – References --> ## References use excellent APA style: <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Very good use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Good use of figure(s) # Excellent use of table(s) # Basic use of feature box(es) # Good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Very good use of quiz(zes) and/or reflection question(s) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section ## Add Wikipedia links <!-- Learning features – External links --> # Basic use of the [[{{PAGENAME}}#External links|External links]] section ## Target an international audience ## Add more links |9= <!-- Social contribution comments... --> # ~5 logged, useful, mostly minor to moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] }} jq6ilhdmi7at7eufawbzyd3vkw8pvyh 2844147 2844146 2026-11-03T09:07:41Z Jtneill 10242 Sign 2844147 wikitext text/x-wiki == Helpful peer reviewed APA 7 article for Hygiene Motivation if you wish to add or remove!! == The concept of hygiene motivation is examined in the study by Curtis, V., & Cairncross, S. (2003), which emphasises that people's hygiene behaviours are primarily driven by the desire to avoid disgust rather than by health education alone. The authors contend that disgust is a potent motivator that can be used in public health campaigns to promote better hygiene practices, particularly in prevention of disease transmission. The study underscores the significance of comprehending emotional drivers like disgust in designing more successful health interventions. Curtis, V., & Cairncross, S. (2003). Effect of washing hands with soap on diarrhoea risk in the community: a systematic review. ''The Lancet Infectious Diseases'', ''3''(5), 275–281. <nowiki>https://doi.org/10.1016/S1473-3099(03)00606-6</nowiki> [[User:U3214564|U3214564]] ([[User talk:U3214564|discuss]] • [[Special:Contributions/U3214564|contribs]]) 08:19, 15 August 2024 (UTC) == Feedback and Suggestions == Hey Jack, I really like your book chapter so far! I found this article from Miller in 2020 which covers motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak. I think this would be an interesting topic to explore and touch on within your book chapter. This article focusses primarily on the role of human behaviour in the COVID-19 outbreak and the speed of the spread across the world, and how human hygiene motivation was associated with the progression. Here is the reference to the article: Gibson Miller, J. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British Journal of Health Psychology'', ''25''(4), 856–864. <nowiki>https://doi.org/10.1111/bjhp.12426</nowiki> Good luck! :) - Alyssia [[User:Alyssia Myers|Alyssia Myers]] ([[User talk:Alyssia Myers|discuss]] • [[Special:Contributions/Alyssia Myers|contribs]]) 11:15, 15 August 2024 (UTC) Hello Jack, it looks like you are hard at work on your page, I've adding another figure to your page, hopefully it'll help you flesh out your article !![https://commons.wikimedia.org/wiki/File:OCD_handwash.jpg this is the image], you could use it to talk about personal hygiene, such as washing one's hands !!! Good luck [[User:Gabriel Geld|Gabriel Geld]] ([[User talk:Gabriel Geld|discuss]] • [[Special:Contributions/Gabriel Geld|contribs]]) 08:43, 16 August 2024 (UTC) <!-- Official topic development feedback --> {{METF/2024 |1= <!-- Title --> # The title and sub-title are correctly worded; casing has been corrected # I've abbreviated the sub-title (same meaning) |2= <!-- Headings --> # Well developed 2-level heading structure. Meaningful headings clearly relate directly to the core topic. # Consider abbreviating the two introductory sections (2 and 3) into a single section; it is OK to retain but keep these sections relatively brief, with embedded links to further information so that the chapter concentrates on psychological science # Remove trailing colons <!-- Alignment with focus questions --> # Good alignment between focus questions and heading structure |3= <!-- Overview--> <!-- Scenario --> # A scenario or case study is presented in a feature box at the start of this section # Add an image to the scenario or case study to help attract reader interest <!-- Description --> # Add a brief, evocative description of the problem/topic <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings # Use sentence casing (Hygiene -> hygiene) # Move quiz into a subsequent section |4= <!-- Key points--> # Partial development of key points for some sections, with limited use of relevant citations <!-- Theory and research --> # Strive for an integrated balance of theory and research, with practical examples # It is unclear whether the best available psychological theory and research has been consulted in the preparation of this plan <!-- Conclusion --> # Conclusion (the most important section): ## Hasn't been developed ## What might the take-home, practical messages be? (What are the answer(s) to the question(s) in the sub-title and/or focus questions?) |5= <!-- Figure --> # A relevant figure is presented, captioned, and cited # The figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Caption citations should refer to Figure rather than figure (APA style) |6= <!-- Learning feature --> <!-- Interwiki links ---> # One use of in-text [[m:Help:Interwiki linking|interwiki links]] for the first mention of key terms to relevant Wikipedia articles and/or to other relevant book chapters <!-- Examples/case studies --> # Promising use of example(s)/case study(ies) <!-- Other --> # Consider including more examples/case studies, quiz question(s), table(s) etc. |7= <!-- References --> <!-- Overall --> # Insufficient <!-- Suggestions --> # Move non-academic / non-peer reviewed sources to External links # Are there any systematic reviews about this topic? |8= <!-- Resources --> # See also ## Not developed # External links ## Not developed |9= <!-- User page --> # Used effectively <!-- Description about self --> # Brief description about self provided – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # Link provided to book chapter |10= <!-- Social contribution --> # At least three different types of contributions with direct link(s) to evidence # The links to evidence can be improved (i.e., be more direct/accurate). For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:53, 18 August 2024 (UTC) == social contribution == hey, I like the case study you have used. linking it to how hygiene decline directly impacts him emotionally and mentally. You could expand on the motivation factors leading to this decline. you could link it to psychological theories and discuss motivational theories like Maslow’s hierarchy of needs or self-determination theory in relation to hygiene maintenance to add to the depth of your chapter. you could also expand on methods to increase hygiene motivation with actionable steps. Great work so far and good luck :) [[User:U3236683|U3236683]] ([[User talk:U3236683|discuss]] • [[Special:Contributions/U3236683|contribs]]) 07:00, 3 October 2024 (UTC) == General tips == Hello, Here are some suggestions to enhance your chapter: First, consider simplifying John’s scenario to make it more relatable. Clearly outline how his hygiene standards declined due to increased stress. This will help readers connect with his situation better. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1.r. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1. [[User:U3219927|U3219927]] ([[User talk:U3219927|discuss]] • [[Special:Contributions/U3219927|contribs]]) 11:35, 6 October 2024 (UTC) == Helpful references == Hi there, I have a suggestion for some references you may like to look at for your topic! '''Motivation Hygiene Theory:''' Brenner, V. C., Carmack, C. W., & Weinstein, M. G. (1971). An empirical test of the motivation-hygiene theory. ''Journal of Accounting Research'', 359-366. '''Application to real world & COVID:''' Gibson Miller, J., Hartman, T. K., Levita, L., Martinez, A. P., Mason, L., McBride, O., ... & Bentall, R. P. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British journal of health psychology'', ''25''(4), 856-864. [[User:Sienna33309|Sienna33309]] ([[User talk:Sienna33309|discuss]] • [[Special:Contributions/Sienna33309|contribs]]) 02:53, 20 August 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] <!-- Alignment with focus questions --> # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # The COVID section could be a subheading |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings # Use open- rather then close-ended focus questions |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad), but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title <!-- Theory and research --> # Promising balance of theory and research <!-- Conclusion --> # Conclusion is well underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) <!-- Tables --> # Excellent use of additional images # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # Only cite sources that you have consulted <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## italicisation ## use dois where available instead of other links ## include hyperlinked dois ## use the simplest, direct link for sources without dois |8= <!-- Resources --> <!-- See also --> # See also ## Basic ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Excellent |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:22, 31 August 2026 (UTC) == Feedback == Hi! I really like your chapter so far and it's quite clear and easy to understand. The only thing I was wondering whilst reading it was whether the sub heading for the health belief model should be closer to the top since the chapter is based off that and also cause it was mentioned quite early on in the chapter. Just a thought, it's very good so far! [[User:U3280499|U3280499]] ([[User talk:U3280499|discuss]] • [[Special:Contributions/U3280499|contribs]]) 08:49, 8 October 2026 (UTC) [[User:U3280499|U3280499]] ([[User talk:U3280499|discuss]] • [[Special:Contributions/U3280499|contribs]]) 08:50, 8 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # The main area(s) for potential improvement: #* more integrative review of the best psychological research about the topic #* quality of written expression #* better citation of sources using APA style #* integrate more strongly with other chapters by using embedded links to related topics <!-- Overall - Edit summaries --> #* Insufficient use [[w:Help:Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FHygiene_motivation&diff=2841655&oldid=2841264 these copyedits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied <!-- Theory – Builds on --> # Builds effectively on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Very good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags), incluidng the year of publication. <!-- Theory – Examples --> # Very good use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Very good review of relevant research # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # More detail about key studies would be ideal # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). <!-- Research – Critical thinking --> # Basic [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Very good integration between the most relevant theory(ies) and the best research # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence |6= <!-- Conclusion comments... --> # Very good summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is very good ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) <!-- Written expression – Grammar --> # Grammar ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – APA style --> # APA style ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – Figures --> ## Figures ### Well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Tables --> ## Tables ### Well captioned ### Use APA style captions (capitalisation) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] because claims are made without citations and the citation year is often missing ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources <!-- Written expression – References --> ## References use excellent APA style: <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Very good use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Good use of figure(s) # Excellent use of table(s) # Basic use of feature box(es) # Good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Very good use of quiz(zes) and/or reflection question(s) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section ## Add Wikipedia links <!-- Learning features – External links --> # Basic use of the [[{{PAGENAME}}#External links|External links]] section ## Target an international audience ## Add more links |9= <!-- Social contribution comments... --> # ~5 logged, useful, mostly minor to moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:07, 3 November 2026 (UTC) 71gzs8ftc31oc5pkkhtbr1sm0h0g3gw 2844148 2844147 2026-11-03T09:10:39Z Jtneill 10242 Removed No use of quiz(zes) and/or reflection question(s) 2844148 wikitext text/x-wiki == Helpful peer reviewed APA 7 article for Hygiene Motivation if you wish to add or remove!! == The concept of hygiene motivation is examined in the study by Curtis, V., & Cairncross, S. (2003), which emphasises that people's hygiene behaviours are primarily driven by the desire to avoid disgust rather than by health education alone. The authors contend that disgust is a potent motivator that can be used in public health campaigns to promote better hygiene practices, particularly in prevention of disease transmission. The study underscores the significance of comprehending emotional drivers like disgust in designing more successful health interventions. Curtis, V., & Cairncross, S. (2003). Effect of washing hands with soap on diarrhoea risk in the community: a systematic review. ''The Lancet Infectious Diseases'', ''3''(5), 275–281. <nowiki>https://doi.org/10.1016/S1473-3099(03)00606-6</nowiki> [[User:U3214564|U3214564]] ([[User talk:U3214564|discuss]] • [[Special:Contributions/U3214564|contribs]]) 08:19, 15 August 2024 (UTC) == Feedback and Suggestions == Hey Jack, I really like your book chapter so far! I found this article from Miller in 2020 which covers motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak. I think this would be an interesting topic to explore and touch on within your book chapter. This article focusses primarily on the role of human behaviour in the COVID-19 outbreak and the speed of the spread across the world, and how human hygiene motivation was associated with the progression. Here is the reference to the article: Gibson Miller, J. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British Journal of Health Psychology'', ''25''(4), 856–864. <nowiki>https://doi.org/10.1111/bjhp.12426</nowiki> Good luck! :) - Alyssia [[User:Alyssia Myers|Alyssia Myers]] ([[User talk:Alyssia Myers|discuss]] • [[Special:Contributions/Alyssia Myers|contribs]]) 11:15, 15 August 2024 (UTC) Hello Jack, it looks like you are hard at work on your page, I've adding another figure to your page, hopefully it'll help you flesh out your article !![https://commons.wikimedia.org/wiki/File:OCD_handwash.jpg this is the image], you could use it to talk about personal hygiene, such as washing one's hands !!! Good luck [[User:Gabriel Geld|Gabriel Geld]] ([[User talk:Gabriel Geld|discuss]] • [[Special:Contributions/Gabriel Geld|contribs]]) 08:43, 16 August 2024 (UTC) <!-- Official topic development feedback --> {{METF/2024 |1= <!-- Title --> # The title and sub-title are correctly worded; casing has been corrected # I've abbreviated the sub-title (same meaning) |2= <!-- Headings --> # Well developed 2-level heading structure. Meaningful headings clearly relate directly to the core topic. # Consider abbreviating the two introductory sections (2 and 3) into a single section; it is OK to retain but keep these sections relatively brief, with embedded links to further information so that the chapter concentrates on psychological science # Remove trailing colons <!-- Alignment with focus questions --> # Good alignment between focus questions and heading structure |3= <!-- Overview--> <!-- Scenario --> # A scenario or case study is presented in a feature box at the start of this section # Add an image to the scenario or case study to help attract reader interest <!-- Description --> # Add a brief, evocative description of the problem/topic <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings # Use sentence casing (Hygiene -> hygiene) # Move quiz into a subsequent section |4= <!-- Key points--> # Partial development of key points for some sections, with limited use of relevant citations <!-- Theory and research --> # Strive for an integrated balance of theory and research, with practical examples # It is unclear whether the best available psychological theory and research has been consulted in the preparation of this plan <!-- Conclusion --> # Conclusion (the most important section): ## Hasn't been developed ## What might the take-home, practical messages be? (What are the answer(s) to the question(s) in the sub-title and/or focus questions?) |5= <!-- Figure --> # A relevant figure is presented, captioned, and cited # The figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Caption citations should refer to Figure rather than figure (APA style) |6= <!-- Learning feature --> <!-- Interwiki links ---> # One use of in-text [[m:Help:Interwiki linking|interwiki links]] for the first mention of key terms to relevant Wikipedia articles and/or to other relevant book chapters <!-- Examples/case studies --> # Promising use of example(s)/case study(ies) <!-- Other --> # Consider including more examples/case studies, quiz question(s), table(s) etc. |7= <!-- References --> <!-- Overall --> # Insufficient <!-- Suggestions --> # Move non-academic / non-peer reviewed sources to External links # Are there any systematic reviews about this topic? |8= <!-- Resources --> # See also ## Not developed # External links ## Not developed |9= <!-- User page --> # Used effectively <!-- Description about self --> # Brief description about self provided – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # Link provided to book chapter |10= <!-- Social contribution --> # At least three different types of contributions with direct link(s) to evidence # The links to evidence can be improved (i.e., be more direct/accurate). For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:53, 18 August 2024 (UTC) == social contribution == hey, I like the case study you have used. linking it to how hygiene decline directly impacts him emotionally and mentally. You could expand on the motivation factors leading to this decline. you could link it to psychological theories and discuss motivational theories like Maslow’s hierarchy of needs or self-determination theory in relation to hygiene maintenance to add to the depth of your chapter. you could also expand on methods to increase hygiene motivation with actionable steps. Great work so far and good luck :) [[User:U3236683|U3236683]] ([[User talk:U3236683|discuss]] • [[Special:Contributions/U3236683|contribs]]) 07:00, 3 October 2024 (UTC) == General tips == Hello, Here are some suggestions to enhance your chapter: First, consider simplifying John’s scenario to make it more relatable. Clearly outline how his hygiene standards declined due to increased stress. This will help readers connect with his situation better. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1.r. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1. [[User:U3219927|U3219927]] ([[User talk:U3219927|discuss]] • [[Special:Contributions/U3219927|contribs]]) 11:35, 6 October 2024 (UTC) == Helpful references == Hi there, I have a suggestion for some references you may like to look at for your topic! '''Motivation Hygiene Theory:''' Brenner, V. C., Carmack, C. W., & Weinstein, M. G. (1971). An empirical test of the motivation-hygiene theory. ''Journal of Accounting Research'', 359-366. '''Application to real world & COVID:''' Gibson Miller, J., Hartman, T. K., Levita, L., Martinez, A. P., Mason, L., McBride, O., ... & Bentall, R. P. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British journal of health psychology'', ''25''(4), 856-864. [[User:Sienna33309|Sienna33309]] ([[User talk:Sienna33309|discuss]] • [[Special:Contributions/Sienna33309|contribs]]) 02:53, 20 August 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. # Clear [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] <!-- Alignment with focus questions --> # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # The COVID section could be a subheading |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings # Use open- rather then close-ended focus questions |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad), but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title <!-- Theory and research --> # Promising balance of theory and research <!-- Conclusion --> # Conclusion is well underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) <!-- Tables --> # Excellent use of additional images # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # Only cite sources that you have consulted <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## italicisation ## use dois where available instead of other links ## include hyperlinked dois ## use the simplest, direct link for sources without dois |8= <!-- Resources --> <!-- See also --> # See also ## Basic ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Excellent |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:22, 31 August 2026 (UTC) == Feedback == Hi! I really like your chapter so far and it's quite clear and easy to understand. The only thing I was wondering whilst reading it was whether the sub heading for the health belief model should be closer to the top since the chapter is based off that and also cause it was mentioned quite early on in the chapter. Just a thought, it's very good so far! [[User:U3280499|U3280499]] ([[User talk:U3280499|discuss]] • [[Special:Contributions/U3280499|contribs]]) 08:49, 8 October 2026 (UTC) [[User:U3280499|U3280499]] ([[User talk:U3280499|discuss]] • [[Special:Contributions/U3280499|contribs]]) 08:50, 8 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # The main area(s) for potential improvement: #* more integrative review of the best psychological research about the topic #* quality of written expression #* better citation of sources using APA style #* integrate more strongly with other chapters by using embedded links to related topics <!-- Overall - Edit summaries --> #* Insufficient use [[w:Help:Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FHygiene_motivation&diff=2841655&oldid=2841264 these copyedits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied <!-- Theory – Builds on --> # Builds effectively on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Very good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags), incluidng the year of publication. <!-- Theory – Examples --> # Very good use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Very good review of relevant research # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # More detail about key studies would be ideal # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). <!-- Research – Critical thinking --> # Basic [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Very good integration between the most relevant theory(ies) and the best research # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence |6= <!-- Conclusion comments... --> # Very good summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is very good ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) <!-- Written expression – Grammar --> # Grammar ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – APA style --> # APA style ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – Figures --> ## Figures ### Well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Tables --> ## Tables ### Well captioned ### Use APA style captions (capitalisation) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] because claims are made without citations and the citation year is often missing ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources <!-- Written expression – References --> ## References use excellent APA style: <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Very good use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Good use of figure(s) # Excellent use of table(s) # Basic use of feature box(es) # Good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Very good use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section ## Add Wikipedia links <!-- Learning features – External links --> # Basic use of the [[{{PAGENAME}}#External links|External links]] section ## Target an international audience ## Add more links |9= <!-- Social contribution comments... --> # ~5 logged, useful, mostly minor to moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:07, 3 November 2026 (UTC) eurkb9fholddz4zrdk9e15x2is0q7ae 1 312568 2844082 2843304 2026-11-03T02:49:23Z Jtneill 10242 2844082 wikitext text/x-wiki == Acknowledgement of assistance == Generative AI assisted with drafting, editing and formatting. The author must review and verify the chapter, sources and final content before submission. == Helpful Resource Suggestions == Hi, I think you have an interesting chapter. I have found some articles that may be interesting to add to your chapter. https://doi.org/10.1111/risa.12179 - The Affect Heuristic Causes Biased Decisions https://doi.org/10.1037/0278-6133.24.4.S35 - Affect (Heuristic), Risk, and Decision Making. [[User:Kiara Stone (UC)|Kiara Stone (UC)]] ([[User talk:Kiara Stone (UC)|discuss]] • [[Special:Contributions/Kiara Stone (UC)|contribs]]) 07:19, 18 October 2024 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Consider integrating theory and research rather than having separate sections (see book chapter marking criteria) # Check grammar for the last heading before the Conclusion <!-- Alignment with focus questions --> # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Good <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # Move the scenario or case study into a feature box to the start of this section to help engage reader interest (fixed) # Add an image to the scenario to help attract reader interest # Make the scenario more practical or down-to-earth; it is currently too abstract <!-- Description --> # A promising description of the problem/topic is planned or presented # What is the key theory and research that will be covered? <!-- Focus questions --> # Reasonably good alignment between focus questions and heading structure, but consider closer alignment # Consider revising FQs 2 and 3 because they are currently very general and not integrated. Perhaps they aren't needed because theory and research could be embedded in other sections. Or as you get to know the theory and research better, consider providing more specific questions. |4= <!-- Key points--> <!-- Overall --> # Key points are well developed for each section # Briefly explain what a heuristic is # Briefly explain what affect is # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) <!-- Theory and research --> # Promising balance of theory and research <!-- GenAI ---> # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # A relevant figure is planned but not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # A quiz question is planned but not implemented; see [[Help:Quiz]] for how to # Consider revising the quiz question to be more practical and less abstract <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Basic <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## remove bullet points (fixed) ## add hanging indent (fixed) ## proofreading needed (e.g., Finucane doi link not working) ## alphabetical order ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] |8= <!-- Resources --> <!-- See also --> # See also ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ## The motivation and emotion book link has been removed because it is too general <!-- External links --> # External links ## Basic ## One of two required external links provided ## The emotion link has been removed because it is too general |9= <!-- User page --> # Basic # Poorly formatted - see skills taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Is this page based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) # Add link to book chapter |10= <!-- Social contribution --> # One out of three types of contributions made with direct link(s) to evidence. 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On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan Showa era *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】. SoraNews24. 17 May 2025. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] spbaophd0oc5t5u8mpbjp92ubm54jbr 2844006 2844004 2026-11-02T19:19:00Z James500 297601 /* */ Add 2844006 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan Showa era *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] fdlcbf8vg8vwo16ajigrxxzwl8e558i 2844012 2844006 2026-11-02T20:13:38Z James500 297601 /* */ Add 2844012 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan Showa era *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. 1960s *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] 10mnuruxmt0rte42qnvw919iu9c6bms 2844013 2844012 2026-11-02T20:16:29Z James500 297601 /* */ Add 2844013 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan Showa era *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. 1960s *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] l7dcryxbtqr62objd2al1rk1cdvs250 2844014 2844013 2026-11-02T20:24:51Z James500 297601 /* */ Add 2844014 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan Showa era *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] s20elngu9mj740swwz7ja6v66eyzlhl 2844066 2844014 2026-11-02T23:22:42Z James500 297601 /* */ Add 2844066 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan Showa era *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] 2h5ppnp86esbb4ni2zx8ktbwdpj89tt 2844083 2844066 2026-11-03T02:56:28Z James500 297601 /* */ Add 2844083 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] mcrsvqrq61794kyknm5xcoizgi210p6 2844100 2844083 2026-11-03T04:18:04Z James500 297601 /* */ Add 2844100 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] s9v2j97qkn4ochr5i34adg67oty6hwn 2844101 2844100 2026-11-03T04:20:24Z James500 297601 /* */ Add 2844101 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] oopn4of452kb1p04paueg5dp1wd0k1e 2844102 2844101 2026-11-03T04:23:36Z James500 297601 /* */ Add 2844102 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] 1owhoknhgx9y1xzegw3vsprswrlonf0 2844105 2844102 2026-11-03T04:42:34Z James500 297601 /* */ Add 2844105 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] ikl3q3kehy70ghhydscjkeksq0g13ag 2844106 2844105 2026-11-03T04:45:58Z James500 297601 /* */ Add 2844106 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Heisei era *Jessica Speed. [https://www.japantimes.co.jp/life/2025/12/06/lifestyle/heisei-retro-gen-z-fashion-cafe/ ‘Messy, blingy, over the top’: Heisei Era charms Japan’s youth]. The Japan Times. 6 December 2025. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] ke1asdsy1u6ct6zwikn9ofg2rq362md 2844107 2844106 2026-11-03T04:48:58Z James500 297601 /* */ Add 2844107 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. *[https://www.nikkei.com/nkd/company/article/?DisplayType=1&ng=DGXZQOUC165OG0W5A211C2000000&scode=2503 昭和レトロ広告、逆にナウい　過去作品にオマージュも]. Nikkei. 29 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Heisei era *Jessica Speed. [https://www.japantimes.co.jp/life/2025/12/06/lifestyle/heisei-retro-gen-z-fashion-cafe/ ‘Messy, blingy, over the top’: Heisei Era charms Japan’s youth]. The Japan Times. 6 December 2025. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] hbc6y2qrbo3bc6eib9y3mfx3e6h4x31 2844108 2844107 2026-11-03T05:47:43Z James500 297601 /* */ Add 2844108 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. *Brendan O'Neill. [http://news.bbc.co.uk/1/hi/magazine/3465231.stm Why nostalgia's not what it used to be]. BBC News. Magazine. [http://news.bbc.co.uk/2/hi/uk_news/magazine/3465231.stm] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. *[https://www.nikkei.com/nkd/company/article/?DisplayType=1&ng=DGXZQOUC165OG0W5A211C2000000&scode=2503 昭和レトロ広告、逆にナウい　過去作品にオマージュも]. Nikkei. 29 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Heisei era *Jessica Speed. [https://www.japantimes.co.jp/life/2025/12/06/lifestyle/heisei-retro-gen-z-fashion-cafe/ ‘Messy, blingy, over the top’: Heisei Era charms Japan’s youth]. The Japan Times. 6 December 2025. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] bncqrj5jbnqiceetuq0pev6vvdopeb0 2844109 2844108 2026-11-03T05:48:34Z James500 297601 /* */ Add 2844109 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. *Brendan O'Neill. [http://news.bbc.co.uk/1/hi/magazine/3465231.stm Why nostalgia's not what it used to be]. BBC News. Magazine. [http://news.bbc.co.uk/2/hi/uk_news/magazine/3465231.stm] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] *[https://fortune.com/2019/09/11/eighties-on-television-glow-stranger-things-goldbergs-ahs/ Behind the ’80s TV Comeback: From ‘Stranger Things’ to ‘GLOW’]. Fortune. 11 September 2019. Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. *[https://www.nikkei.com/nkd/company/article/?DisplayType=1&ng=DGXZQOUC165OG0W5A211C2000000&scode=2503 昭和レトロ広告、逆にナウい　過去作品にオマージュも]. Nikkei. 29 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Heisei era *Jessica Speed. [https://www.japantimes.co.jp/life/2025/12/06/lifestyle/heisei-retro-gen-z-fashion-cafe/ ‘Messy, blingy, over the top’: Heisei Era charms Japan’s youth]. The Japan Times. 6 December 2025. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] 4lqgqbmt64abquj6cvrkps2d2dub0ch 2844110 2844109 2026-11-03T06:02:14Z James500 297601 /* */ Add 2844110 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. *Brendan O'Neill. [http://news.bbc.co.uk/1/hi/magazine/3465231.stm Why nostalgia's not what it used to be]. BBC News. Magazine. [http://news.bbc.co.uk/2/hi/uk_news/magazine/3465231.stm] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] *[https://fortune.com/2019/09/11/eighties-on-television-glow-stranger-things-goldbergs-ahs/ Behind the ’80s TV Comeback: From ‘Stranger Things’ to ‘GLOW’]. Fortune. 11 September 2019. Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://www.yomiuri.co.jp/life/20250913-OYT8T50023/ ［日曜の朝に］昭和レトロに魅せられて]. Yomiuri Shimbun. 14 September 2025. *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. *[https://www.nikkei.com/nkd/company/article/?DisplayType=1&ng=DGXZQOUC165OG0W5A211C2000000&scode=2503 昭和レトロ広告、逆にナウい　過去作品にオマージュも]. Nikkei. 29 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Heisei era *Jessica Speed. [https://www.japantimes.co.jp/life/2025/12/06/lifestyle/heisei-retro-gen-z-fashion-cafe/ ‘Messy, blingy, over the top’: Heisei Era charms Japan’s youth]. The Japan Times. 6 December 2025. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] fgcufkq936zytmmpidq4ekigkd0a068 2844113 2844110 2026-11-03T06:15:12Z James500 297601 /* */ Add 2844113 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. *Brendan O'Neill. [http://news.bbc.co.uk/1/hi/magazine/3465231.stm Why nostalgia's not what it used to be]. BBC News. Magazine. [http://news.bbc.co.uk/2/hi/uk_news/magazine/3465231.stm] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] *[https://fortune.com/2019/09/11/eighties-on-television-glow-stranger-things-goldbergs-ahs/ Behind the ’80s TV Comeback: From ‘Stranger Things’ to ‘GLOW’]. Fortune. 11 September 2019. Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. *[https://www.apa.org/monitor/2026/11-12/gen-z-nostalgia-identity-meaning.html Why Gen Z is nostalgic for a past they never lived]. Monitor on Psychology. November/December 2026. From American Psychological Association. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://www.yomiuri.co.jp/life/20250913-OYT8T50023/ ［日曜の朝に］昭和レトロに魅せられて]. Yomiuri Shimbun. 14 September 2025. *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. *[https://www.nikkei.com/nkd/company/article/?DisplayType=1&ng=DGXZQOUC165OG0W5A211C2000000&scode=2503 昭和レトロ広告、逆にナウい　過去作品にオマージュも]. Nikkei. 29 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Heisei era *Jessica Speed. [https://www.japantimes.co.jp/life/2025/12/06/lifestyle/heisei-retro-gen-z-fashion-cafe/ ‘Messy, blingy, over the top’: Heisei Era charms Japan’s youth]. The Japan Times. 6 December 2025. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] g0umccza3wb1qyfc7ervwstzeb2tfuj 2844120 2844113 2026-11-03T06:49:17Z James500 297601 /* */ Add 2844120 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. *Brendan O'Neill. [http://news.bbc.co.uk/1/hi/magazine/3465231.stm Why nostalgia's not what it used to be]. BBC News. Magazine. [http://news.bbc.co.uk/2/hi/uk_news/magazine/3465231.stm] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] Decade nostalgia *Rod Liddle. [https://www.thetimes.com/uk/politics/article/rolling-back-the-years-qg8fs6sh06d Rolling back the years]. The Times. 10 July 2004. For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] *[https://fortune.com/2019/09/11/eighties-on-television-glow-stranger-things-goldbergs-ahs/ Behind the ’80s TV Comeback: From ‘Stranger Things’ to ‘GLOW’]. Fortune. 11 September 2019. Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. *[https://www.apa.org/monitor/2026/11-12/gen-z-nostalgia-identity-meaning.html Why Gen Z is nostalgic for a past they never lived]. Monitor on Psychology. November/December 2026. From American Psychological Association. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://www.yomiuri.co.jp/life/20250913-OYT8T50023/ ［日曜の朝に］昭和レトロに魅せられて]. Yomiuri Shimbun. 14 September 2025. *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. *[https://www.nikkei.com/nkd/company/article/?DisplayType=1&ng=DGXZQOUC165OG0W5A211C2000000&scode=2503 昭和レトロ広告、逆にナウい　過去作品にオマージュも]. Nikkei. 29 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Heisei era *Jessica Speed. [https://www.japantimes.co.jp/life/2025/12/06/lifestyle/heisei-retro-gen-z-fashion-cafe/ ‘Messy, blingy, over the top’: Heisei Era charms Japan’s youth]. The Japan Times. 6 December 2025. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. [[Category:Culture]] 9hd3965o4ledimejq4avm6heaehgc7y 2844122 2844120 2026-11-03T06:57:38Z James500 297601 /* */ Add 2844122 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. *Brendan O'Neill. [http://news.bbc.co.uk/1/hi/magazine/3465231.stm Why nostalgia's not what it used to be]. BBC News. Magazine. [http://news.bbc.co.uk/2/hi/uk_news/magazine/3465231.stm] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] Decade nostalgia *Rod Liddle. [https://www.thetimes.com/uk/politics/article/rolling-back-the-years-qg8fs6sh06d Rolling back the years]. The Times. 10 July 2004. For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] *[https://fortune.com/2019/09/11/eighties-on-television-glow-stranger-things-goldbergs-ahs/ Behind the ’80s TV Comeback: From ‘Stranger Things’ to ‘GLOW’]. Fortune. 11 September 2019. Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. *[https://www.apa.org/monitor/2026/11-12/gen-z-nostalgia-identity-meaning.html Why Gen Z is nostalgic for a past they never lived]. Monitor on Psychology. November/December 2026. From American Psychological Association. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://www.yomiuri.co.jp/life/20250913-OYT8T50023/ ［日曜の朝に］昭和レトロに魅せられて]. Yomiuri Shimbun. 14 September 2025. *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. *[https://www.nikkei.com/nkd/company/article/?DisplayType=1&ng=DGXZQOUC165OG0W5A211C2000000&scode=2503 昭和レトロ広告、逆にナウい　過去作品にオマージュも]. Nikkei. 29 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Heisei era *Jessica Speed. [https://www.japantimes.co.jp/life/2025/12/06/lifestyle/heisei-retro-gen-z-fashion-cafe/ ‘Messy, blingy, over the top’: Heisei Era charms Japan’s youth]. The Japan Times. 6 December 2025. [[Category:Culture]] 6s4joc7ugd7s9dcb9t8rgd13vtu4um6 2844141 2844122 2026-11-03T08:53:27Z James500 297601 /* */ Add 2844141 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of nostalgia and retro. *The Routledge Handbook of Nostalgia [https://books.google.co.uk/books?id=EYgYEQAAQBAJ&pg=PA1977#v=onepage&q&f=false] *David Berry. On Nostalgia. 2020. [https://books.google.co.uk/books?id=U9TXDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Exploring Nostalgia: Sad, Bad, Mad and Sweet. 2016. [https://books.google.co.uk/books?id=4dweEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Illbruck. Nostalgia: Origins and Ends of an Unenlightened Disease. 2012. [https://books.google.co.uk/books?id=tlmihfXbud8C&pg=PP1#v=onepage&q&f=false] *McMahon, "Nostalgia".Kammen and Prendergast (eds). Encyclopedia of Local History. 2000. [https://books.google.co.uk/books?id=nf_M9vqUMLYC&pg=PA348#v=onepage&q&f=false p 348]. *Boym. The Future Of Nostalgia. 2001. [https://books.google.com/books?id=WrvtAAAAMAAJ] *Cassin. Nostalgia: When are We Ever at Home? 2016. [https://books.google.co.uk/books?id=yhNPyQEACAAJ] *Jacobsen (ed). Nostalgia Now: Cross-Disciplinary Perspectives on the Past in the Present. 2020. [https://books.google.co.uk/books?id=eaPSDwAAQBAJ&pg=PA1983#v=onepage&q&f=false] *Jacobsen (ed). Intimations of Nostalgia: Multidisciplinary Explorations of an Enduring Emotion. 2022. [https://books.google.co.uk/books?id=LpxKEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Gandini. Zeitgeist Nostalgia: On Populism, Work and the ‘Good Life’. 2020. [https://books.google.co.uk/books?id=hrICEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Howard, "Nostalgia" (2012) 72 Analysis 641 [https://www.jstor.org/stable/23359115] *Wilson, "Remember When: A Consideration of the Concept of Nostalgia" (1999) 56 ETC: A Review of General Semantics 296 [https://www.jstor.org/stable/42705763] *Becker, "The Meanings of Nostalgia" (2018) 57 History and Theory 234 [https://www.jstor.org/stable/26650760] *Chrostowska, "Consumed by Nostalgia?" (2010) 39 SubStance 52 [https://www.jstor.org/stable/40801075] *Cross. Consumed Nostalgia: Memory in the Age of Fast Capitalism. 2015. [https://books.google.co.uk/books?id=EyVBCgAAQBAJ&pg=PA1848#v=onepage&q&f=false] *[https://www.sfchronicle.com/entertainment/article/sf-retro-nostalgia-trends-18638368.php Retro revival: How nostalgia shapes modern culture]. San Francisco Chronicle. 20 December 2024. *Brendan O'Neill. [http://news.bbc.co.uk/1/hi/magazine/3465231.stm Why nostalgia's not what it used to be]. BBC News. Magazine. [http://news.bbc.co.uk/2/hi/uk_news/magazine/3465231.stm] History *Shaw and Chase. The Imagined Past: History and Nostalgia. 1989. [https://books.google.co.uk/books?id=vNNRAQAAIAAJ&pg=PP1#v=onepage&q&f=false] *Becker. Yesterday: A New History of Nostalgia. 2023. [https://books.google.co.uk/books?id=OGnQEAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.newyorker.com/magazine/2023/11/27/yesterday-a-new-history-of-nostalgia-tobias-becker-book-review] [https://www.washingtonpost.com/books/2024/11/01/yesterday-tobias-becker-nostalgia-agnes-arnold-forster-invention-prehistory-stefanos-geroulanos/] *Woods. Rule, Nostalgia: A Backwards History of Britain. 2023. [https://books.google.co.uk/books?id=pmw9EAAAQBAJ&pg=PA1891#v=onepage&q&f=false] *Arnold-Foster. Nostalgia: A History of a Dangerous Emotion. 2024. [https://books.google.co.uk/books?id=11jOEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lyon and Walsham. Nostalgia in the Early Modern World: Memory, Temporality, and Emotion. 2023. [https://books.google.co.uk/books?id=j2i5EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Austin. Nostalgia in Transition, 1780-1917. 2007. [https://books.google.co.uk/books?id=Gl_gVlQdjfQC&pg=PP1#v=onepage&q&f=false] *Landwehr, "Nostalgia and the Turbulence of the Times" (2018) 57 History and Theory 251 [https://www.jstor.org/stable/26650761] *Brandt. "A Short Natural History of Nostalgia". The Atlantic. December 1978. p 58. [https://www.theatlantic.com/magazine/archive/1978/12/a-short-natural-history-of-nostalgia/662384/] *"Local history as productive nostalgia? Change, continuity and sense of place in rural England" (2017) 18 Social & Cultural Geography 466 [https://www.tandfonline.com/doi/abs/10.1080/14649365.2016.1189591] During the Victorian period *Colley. Nostalgia and Recollection in Victorian Culture. 1998. [https://books.google.co.uk/books?id=uaGHDAAAQBAJ&pg=PP1#v=onepage&q&f=false] During the 1970s and 1980s *Dwyer. Back to the Fifties: Nostalgia, Hollywood Film, and Popular Music of the Seventies and Eighties. 2015. [https://books.google.co.uk/books?id=FMj9CAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Alpers. Happy Days: Images of the Pre-Sixties Past in Seventies America. 2024. [https://books.google.co.uk/books?id=Hw3zEAAAQBAJ&pg=PA1955#v=onepage&q&f=false] *Charles Foley. "Won Ton Kong" in "Nostalgia". vol 98. [[w:The Bulletin (Australian periodical)|The Bulletin]]. No 4996: 28 February 1976. [https://nla.gov.au/nla.obj-1447552086/view?sectionId=nla.obj-1649882164&partId=nla.obj-1447711387#page/n78/mode/1up/ p 79]. *John Culhane, "Nostalgia", Newsweek, [https://books.google.co.uk/books?id=PeDjAAAAMAAJ vol 76], 28 December 1970, pp 34 to 38. ("The Old in Vogue"). *"Everybody's Just Wild About . . . Nostalgia. Life. 19 February 1971. [https://books.google.co.uk/books?id=3lMEAAAAMBAJ&pg=PP1#v=onepage&q&f=false] *"Born Too Late? A new generation of rockers gathers at the Lyceum". Time Out. (London's Living Guide). No 287: 12 to 18 September 1975. [Cover: Iain Chambers, Popular Culture: The Metropolitan Experience, Meuthen, 1986, Routledge, 1988, 1990, p 171.] *[https://trove.nla.gov.au/newspaper/article/261562615 "Quality in nostalgia"]. The Australian Jewish News. 4 October 1985. p 33. Decade nostalgia *Rod Liddle. [https://www.thetimes.com/uk/politics/article/rolling-back-the-years-qg8fs6sh06d Rolling back the years]. The Times. 10 July 2004. For the 1980s *Reliving the '80s: Nostalgic implementation of the '80s pop music in the media. University of Nevada. 2006. Thesis. [https://digitalscholarship.unlv.edu/rtds/1946/] *[https://fortune.com/2019/09/11/eighties-on-television-glow-stranger-things-goldbergs-ahs/ Behind the ’80s TV Comeback: From ‘Stranger Things’ to ‘GLOW’]. Fortune. 11 September 2019. Geography *Bonnett. The Geography of Nostalgia: Global and Local Perspectives on Modernity and Loss. 2016. [https://books.google.co.uk/books?id=ceZWCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Anthropology *Angé and Berliner. Anthropology and Nostalgia. 2015. [https://books.google.co.uk/books?id=YCQfAwAAQBAJ&pg=PR3#v=onepage&q&f=false] Sociology *Fred Davis. Yearning for Yesterday: A Sociology of Nostalgia. 1979. [https://books.google.com/books?id=3zt-AAAAMAAJ] *DaSilva and Faught, "Nostalgia: A sphere and process of contemporary ideology" (1982) 5 Qualitative Sociology 47 [https://link.springer.com/article/10.1007/BF01006418] Media *Niemeyer. Media and Nostalgia: Yearning for the Past, Present and Future. 2014. [https://books.google.co.uk/books?id=7SKvAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Lizardi. Mediated Nostalgia: Individual Memory and Contemporary Mass Media. 2015. [https://books.google.co.uk/books?id=fKeDBQAAQBAJ&pg=PP1#v=onepage&q&f=false] Film and television *Leggatt. Was It Yesterday?: Nostalgia in Contemporary Film and Television. 2021. [https://books.google.co.uk/books?id=wZf1DwAAQBAJ&pg=PA1962#v=onepage&q&f=false] *Sperb. Flickers of Film: Nostalgia in the Time of Digital Cinema. 2016. [https://books.google.co.uk/books?id=o9D-CgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Sprengler. Screening Nostalgia: Populuxe Props and Technicolor Aesthetics in Contemporary American Film. 2009. [https://books.google.co.uk/books?id=jCkToFfwOrsC&pg=PP1#v=onepage&q&f=false] *Furnish. Nostalgia in Jewish-American Theatre and Film, 1979-2004. 2005. [https://books.google.co.uk/books?id=treeol9deZ8C&pg=PP1#v=onepage&q&f=false] *Czach and Loiselle. Cinema of Pain: On Quebec's Nostalgic Screen. 2020. [https://books.google.co.uk/books?id=h-nyDwAAQBAJ&pg=PA1971#v=onepage&q&f=false] *Pallister (ed). Netflix Nostalgia: Streaming the Past on Demand. 2019. [https://books.google.co.uk/books?id=9DaQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Armbruster. Watching Nostalgia: An Analysis of Nostalgic Television Fiction and its Reception. 2016. [https://books.google.co.uk/books?id=EidTDQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Bevan. The Aesthetics of Nostalgia TV: Production Design and the Boomer Era. 2019. [https://books.google.co.uk/books?id=5J98DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Radio and television *Punch. 3 December 1969. [https://books.google.co.uk/books?id=oXAgAQAAMAAJ] **"Nostalgia note". Evening Times. 3 December 1969. [https://books.google.co.uk/books?id=4Sw-AAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. Literature *Łaszkiewicz, Maszewski and Partyka. Dwelling in Days Foregone: Nostalgia in American Literature and Culture. 2016. [https://books.google.co.uk/books?id=WTP5DAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Ladino. Reclaiming Nostalgia: Longing for Nature in American Literature. 2012. [https://books.google.co.uk/books?id=3uGMgcG12V0C&pg=PA1#v=onepage&q&f=false] *Trilling. The Aesthetics of Nostalgia: Historical Representation in Old English Verse. 2009. [https://books.google.co.uk/books?id=HALmDQAAQBAJ&pg=PP1#v=onepage&q&f=false] Marketing *Pichierri. Nostalgia Marketing: Rekindling the Past to Influence Consumer Choices. 2023. [https://books.google.co.uk/books?id=ylazEAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Tina Kießling. Nostalgie und Retro-Trends als Marketingchance. 2012. [https://books.google.co.uk/books?id=GHsQmsQWfhEC&pg=PP1#v=onepage&q&f=false] *Moriarty and McGann, "Nostalgia and Consumer Sentiment" (1983) 60 Journalism Quarterly 81 [https://journals.sagepub.com/doi/abs/10.1177/107769908306000113] Nostalgia industry *"Selling Nostalgia Is Big Business". Reading Eagle. 11 March 1962. [https://books.google.co.uk/books?id=2BMrAAAAIBAJ&pg=PA14#v=onepage&q&f=false p 14]. From Associated Press. Cf. [https://books.google.co.uk/books?id=fKEtAAAAIBAJ&pg=PA18#v=onepage&q&f=false][https://books.google.co.uk/books?id=8BYfAAAAIBAJ&pg=PA9#v=onepage&q&f=false] *Hal Boyle. "Hobbyist Saloonkeeper Is Relectant Success". Prescott Evening Courier. 11 August 1959. [https://books.google.co.uk/books?id=gy1PAAAAIBAJ&pg=PA3#v=onepage&q&f=false p 4]. From Associated Press. Cf [https://books.google.co.uk/books?id=dt9NAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=rSATAAAAIBAJ&pg=PA12#v=onepage&q&f=false]. Politics *Quill. Nostalgia and Political Theory. 2024. [https://books.google.co.uk/books?id=9aroEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Tanner. The Hours Have Lost Their Clock: The Politics of Nostalgia. 2021. [https://books.google.co.uk/books?id=d9ATEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Bonnett. Left in the Past: Radicalism and the Politics of Nostalgia. 2010. [https://books.google.co.uk/books?id=e8WoAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Campanella and Dassù. Anglo Nostalgia: The Politics of Emotion in a Fractured West. 2019. [https://books.google.co.uk/books?id=iaCgDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Childhood *Wesseling (ed). Reinventing Childhood Nostalgia: Books, Toys, and Contemporary Media Culture. 2017. [https://books.google.co.uk/books?id=bu80DwAAQBAJ&pg=PA1#v=onepage&q&f=false] Psychology *Routledge. Nostalgia: A Psychological Resource. 2016. [https://books.google.co.uk/books?id=6dyoCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *"Nostalgia: Past, Present, and Future" (2008) 17 Current Directions in Psychological Science 304 [https://www.jstor.org/stable/20183308] *Daniels, "Nostalgia and Hidden Meaning" (1985) 42 American Imago 371 [https://www.jstor.org/stable/26303822] Medical *Dodham. What Nostalgia Was: War, Empire, and the Time of a Deadly Emotion. [https://books.google.co.uk/books?id=VxNDDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Retro *Guffey. Retro: The Culture of Revival. 2006. [https://books.google.co.uk/books?id=2d8J08XNdKEC&pg=PP1#v=onepage&q&f=false] *Reynolds. Retromania: Pop Culture's Addiction to its Own Past. 2011. [https://books.google.co.uk/books?id=0Hgtu9Zwok0C&pg=PA2001#v=onepage&q&f=false] *Brown. Marketing - The Retro Revolution. 2001. [https://books.google.co.uk/books?id=H5G3rOh-VDkC&pg=PP1#v=onepage&q&f=false] Vicarious nostalgia *"Missing Someone Else's Past". Psychology Today. 30 January 2013. [https://www.psychologytoday.com/gb/blog/longing-for-nostalgia/201301/missing-someone-elses-past] *"Effects of advertising-evoked vicarious nostalgia on brand heritage" (2013) 66 Journal of Business Research 2619 [https://www.sciencedirect.com/science/article/abs/pii/S0148296312001737] *"Reminiscing Other People’s Memories: Conceptualizing and Measuring Vicarious Nostalgia Evoked by Heritage Tourism" (2020) 61 Journal of Travel Research [https://journals.sagepub.com/doi/10.1177/0047287520969904] *"Antecedents of tourism vicarious nostalgia". Tourism, Change and the Global South. [https://books.google.co.uk/books?id=jD4qEAAAQBAJ&newbks=1&newbks_redir=0&lpg=PA2006&pg=PA2006#v=onepage&q&f=false Chapter 4]. *[https://www.apa.org/monitor/2026/11-12/gen-z-nostalgia-identity-meaning.html Why Gen Z is nostalgic for a past they never lived]. Monitor on Psychology. November/December 2026. From American Psychological Association. Generation Z *Emily Walker. [https://www.deseret.com/lifestyle/2026/10/10/the-social-effects-of-growing-nostalgia/ Gen Z is feeling nostalgic. Experts say it might be a good thing]. Deseret News. 10 October 2026. *Cynthia Littleton. [https://variety.com/2026/tv/news/gen-z-nostalgia-borrowed-gen-z-millennials-1236806310/ Gen Z Leans Hard into Nostalgia to Spur Music and Content Discovery: Study]. Variety. 10 July 2026. United States *Sayers. The Promise of Nostalgia: Reminiscence, Longing and Hope in Contemporary American Culture. 2020. [https://books.google.co.uk/books?id=BSnJDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Kaiju *Jason Barr. "The Role of Nostalgia". The Kaiju Film: A Critical Study of Cinema's Biggest Monsters. McFarland & Company. 2016. Chapter 7. pp  [https://books.google.co.uk/books?id=eQOQCwAAQBAJ&pg=PA169#v=onepage&q&f=false 169] to 175. Japan *古民家、レトロ、ノスタルジー。[[w:ja:ほっとこうち|Hotkochi]] (ほっとこうち). vol 331. October 2024. [https://hotkochi.co.jp/news/102317/] *[https://www3.nhk.or.jp/nhkworld/news/video/20261013145700349/ Tokyo developers embrace retro buildings]. NHK World-Japan. 13 October 2026. Showa era *[https://www.yomiuri.co.jp/life/20250913-OYT8T50023/ ［日曜の朝に］昭和レトロに魅せられて]. Yomiuri Shimbun. 14 September 2025. *[https://mainichi.jp/articles/20251229/ddl/k08/040/056000c 列島・北から南から 昭和レトロがエモい]. Mainichi Shimbun. 29 December 2025. *Krista Rogers. [https://soranews24.com/2015/09/23/nostalgic-exhibition-in-odaiba-features-rare-and-vintage-character-toys-from-the-1960s-onwards/ Odaiba exhibition features rare & vintage character toys from the 1960s onwards, photos A-OK!] SoraNews24. 23 September 2015. *[https://soranews24.com/2025/05/17/new-retro-museum-in-tokyo-is-like-a-time-portal-back-to-the-showa-era【photos】/ New retro museum in Tokyo is like a time portal back to the Showa era【Photos】]. SoraNews24. 17 May 2025. *[https://soranews24.com/2025/12/11/tokyo-event-lets-you-travel-back-in-time-for-free-to-celebrate-100-years-since-showa-era-start/ Tokyo event lets you travel back in time, for free, to celebrate 100 years since Showa era start]. SoraNews24. 11 December 2025. *[https://www.yomiuri.co.jp/local/tochigi/news/20260106-GYTNT00019/ 昭和レトロが「エモい」…那須塩原で企画展　足踏みミシンや白黒テレビ、茶の間のジオラマも]. Yomiuri Shimbun. 6 January 2026. *[https://www.tokyo-np.co.jp/article/458646 「昭和レトロのエモさ」紹介　那須野が原博物館で企画展　「ノスタルジー」「アナログさ」「おしゃれデザイン」]. Tokyo Shimbun. 26 December 2025. *[https://www.nikkei.com/nkd/company/article/?DisplayType=1&ng=DGXZQOUC165OG0W5A211C2000000&scode=2503 昭和レトロ広告、逆にナウい　過去作品にオマージュも]. Nikkei. 29 December 2025. 1960s *[https://soranews24.com/2018/06/04/cool-history-garage-in-toyotas-massive-car-theme-park-warps-you-back-to-japans-1960s【video】/ Cool History Garage in Toyota’s massive car theme park warps you back to Japan’s 1960s【Video】]. SoraNews24. 4 June 2018. *[https://soranews24.com/2022/11/16/perfectly-recreated-1960s-japanese-apartment-lets-museum-visitors-go-back-to-mid-showa-era【pics】/ Perfectly recreated 1960s Japanese apartment lets museum visitors go back to mid-Showa era【Pics】]. SoraNews24. 16 November 2022. *[https://soranews24.com/2024/09/18/super-budget-friendly-retro-tokyo-hotel-feels-like-having-your-own-1960s-asakusa-apartment/ Super budget-friendly retro Tokyo hotel feels like having your own 1960s Asakusa apartment]. SoraNews24. 18 September 2024. *[https://soranews24.com/2026/09/08/tokyos-red-light-district-cafe-lets-you-dine-inside-a-real-1960s-cabaret/ Tokyo’s red-light district cafe lets you dine inside a real 1960s cabaret]. SoraNews24. 8 September 2026. Heisei era *Jessica Speed. [https://www.japantimes.co.jp/life/2025/12/06/lifestyle/heisei-retro-gen-z-fashion-cafe/ ‘Messy, blingy, over the top’: Heisei Era charms Japan’s youth]. The Japan Times. 6 December 2025. [[Category:Culture]] 3m3y489ujntih41euk2zobh1lj8bfca 0 322987 2844075 2842885 2026-11-03T02:09:20Z Jtneill 10242 2844075 wikitext text/x-wiki {{title|Occupational violence, emotion, and coping:<br>What are the emotional impacts of occupational violence and how can employees cope?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} ;Case study [[File:NounProject Abuse (38107).svg|thumb|204x204px|'''Figure 1'''. Example of verbal abuse.]] Emma is a 21-year-old person of colour who works in aged care. She has been working at a residential facility for the last six months. During a busy evening shift, a resident becomes frustrated by a delay in care and starts shouting at Emma in front of other staff and residents. The resident calls Emma names, including incompetent and racial slurs, points fingers at her, and threatens to make a formal complaint to have her fired. Although the situation is eventually de-escalated by her supervisor, Emma feels shaken for the rest of her shift. {{RoundBoxBottom}} [[wikipedia:Workplace_violence|Occupational violence]] (OV) is a significant challenge in modern workplaces, affecting employees’ physical, emotional, and psychological well-being. While OV can occur in any industry, employees in direct face-to-face roles, such as healthcare, customer service, and aged care, are disproportionately exposed to intimidating, aggressive, and hostile behaviours from clients and patients (Alameddine et al., 2015). OV is not an isolated social issue. It is a complex, systemic problem rooted in broader organisational culture, social power dynamics, and individual psychological health. Exposure to OV can severely affect a worker's emotional stability, decrease job satisfaction, reduce motivation, and trigger severe psychological distress, including burnout and symptoms of post-traumatic stress (Pihl-Thingvad et al., 2019). Emma’s situation shows how OV rarely occurs in a vacuum. Her experience highlights the intersection of operational stressors, such as shift demands and staffing shortages, personal attacks, and demographic vulnerabilities, such as race, gender and age. {{RoundBoxTop}} ; Focus questions * What is occupational violence? * What are the risk factors? * What are the emotional and psychological impacts? * What are the coping mechanisms that can used? * How can organisations prevent it and support employees who are affected? {{RoundBoxBottom}} == Understanding Occupational Violence == [[File:Angry man.svg|thumb|242x242px|'''Figure 2.''' Example of someone being verbally violent.]] OV is defined as an incident in which an employee is abused, threatened, or assaulted in a work setting. OV can be a single incident or repeated behaviour. OV includes physical aggression, [[wikipedia:Verbal_abuse|verbal abuse]], [[wikipedia:Intimidation|intimidation]], [[wikipedia:Microaggression|microaggressions]], and subtle discriminatory behaviours. OV is far from being an unpredictable or isolated interpersonal conflict; it is a systemic workplace safety problem and a significant psychological issue, particularly in public-facing sectors. When organisations frame client aggression as an expected part of the job, they shift the burden of workplace safety onto individual workers, which suppresses incident reporting and leaves staff exposed to chronic workplace stress. {{f}} The Job Demands-Resources (JD-R) model proposed by Bakker and Demerouti (2007) provides a useful framework for understanding the psychological impact of OV. Within this model, OV can be understood as a significant job demand that requires employees to expend emotional and psychological resources to manage stressful or threatening situations. When these demands are not adequately balanced by organisational resources, such as supervisor support and appropriate workplace policies, employees may be at greater risk of psychological harm (Bakker and Demerouti, 2007). Additional research supports this theory, with exposure to workplace violence being associated with anxiety, emotional exhaustion and symptoms of post-traumatic stress (Pihl-Thingvad et al., 2019). These psychological effects may also extend into employees' work experiences, contributing to reduced motivation and job satisfaction and potentially influencing their sense of professional identity. There are two different types of OV, specifically, overt occupational violence and micro-aggression. Overt OV refers to direct and easily identifiable behaviours that threaten an employee’s physical or psychological safety, such as physical and verbal assault, intimidation, or sexual harassment. On the other hand, micro-violence involves more subtle and less obvious behaviours, such as repeated exclusion, dismissive comments, and belittling. While these behaviours may appear less severe when counted as single incidents, repeated exposure can negatively affect an employee’s sense of safety and psychological well-being. Therefore, differentiating between overt and micro forms of OV is important because less visible behaviours may be more difficult to recognise, report, and address within the workplace. {{f}} ==== Forms of occupational violence ==== There are several forms of OV, which range from direct aggression to more subtle behaviours that negatively affect an employee's psychological well-being. Verbal violence involves threats, verbal abuse, yelling, swearing, or intimidating language directed towards an employee. [[wikipedia:Bullying|Bullying]] and harassment generally involve repeated behaviours that intimidate, humiliate, undermine, or otherwise cause psychological harm and stress to an employee. In comparison, [[wikipedia:Discrimination|discrimination]] and [[wikipedia:Social_exclusion|exclusion]] involve unfair or unequal treatment based on characteristics such as race, gender, nationality, or [[wikipedia:Sexual_orientation|sexual orientation]], and may include social isolation, favouritism, or exclusion from workplace opportunities. [[wikipedia:Sexual_harassment|Sexual harassment]] refers to unwanted sexual behaviour, including inappropriate touching, sexual comments, unwanted advances, or other conduct that violates an employee's personal boundaries. Although these forms of OV can differ in how they are expressed, they can all negatively affect an employee's sense of safety, wellbeing, and experience within the workplace. {{f}} ==== What are the risks {{g}} ==== A range of specific workplace and interpersonal factors, including high [[wikipedia:Workload|workloads]], stressful working conditions, poor communication, power imbalances, inadequate supervision, and interactions with aggressive clients, customers, patients, or colleagues, can influence OV {{f}}. Workplace culture also plays an important role in either increasing or reducing these risks {{f}}. For example, workplaces that tolerate aggressive behaviour, discourage reporting, or provide limited management support may allow harmful behaviours to continue (George, 2023). In contrast, supportive leadership, clear behavioural expectations, and effective reporting procedures can help reduce the risk of OV (Mache et al., 2026). Additionally, some employees may also be more vulnerable to discrimination and harassment because of factors such as race, skin colour, cultural background, sexual orientation, or gender {{f}}. These characteristics do not cause OV; rather, discrimination, prejudice, and poor organisational practices can increase an employee's exposure to harmful behaviour {{f}}. {{RoundBoxTop|theme=9}} ;Key points *OV can occur as a single incident or repeated harmful behaviours. *OV include both overt behaviours, such as physical violence and verbal abuse and less visible behaviours, such as microaggression and discrimination. *Workplace conditions, including high workloads and inadequate supervision, can increase employee's exposure to OV. *Discrimination and prejudice may place certain employees at a greater risk of OV. {{RoundBoxBottom}} <quiz display="simple"> {A patient starts shouting and making personal attacks at a worker. What type of occupational violence is this? |type="()"} - Microaggression + Overt verbal violence - Physical assault </quiz> == Emotional and Psychological Impacts == The effects of OV extend beyond the immediate incident; it influences employees' emotional responses, psychological safety, and attitudes towards their work {{f}}. Employees who encounter any form of OV may experience fear, anxiety, anger, humiliation, and emotional exhaustion {{f}}. The severity and duration of these effects can vary depending on the nature of the incident, whether exposure continues, and the support available to the employee {{f}}. ==== Emotional impacts ==== Following exposure to OV, employees may demonstrate a range of acute and persistent psychological and emotional responses, including fear, anxiety, anger, frustration, sadness, humiliation, shame, and emotional exhaustion {{f}}. These emotional outcomes can occur immediately post-incident or persist as chronic psychological trauma, particularly when there is a lack of workplace safety (Brooks et al., 2017). For example, an employee subjected to verbal abuse from a client may experience heightened anxiety during an interaction with that client again, whereas an employee who experiences public humiliation by a colleague may feel profound shame and uneasiness regarding how other colleagues perceive them. Although these responses differ, they can all influence how safe and confident an employee feels in their workplace {{f}}. Research also supports the relationship between workplace violence and negative emotional experiences. According to a systematic review {{ic|of how many studies?}} by Lanctôt and Guay (2014), anger, fear, and other emotional reactions were reported as effects in studies about healthcare workers exposed to workplace violence. Extending these findings, a systematic review and meta-analysis {{ic|of how many studies?}} focusing on female healthcare professionals by Ajuwa et al. (2024) similarly found emotional and psychological effects, specifically identifying elevated levels of anger, fear, shame, anxiety, and psychological stress. Together, these reviews suggest that the emotional consequences of OV extend beyond the immediate experience of being threatened or harmed. Employees may also experience ongoing anxiety about returning to the workplace or encountering the person responsible. The stress-buffering hypothesis by Cohen and Wills (1985) provides further explanation for how organisational support may influence employees' emotional responses to OV. This hypothesis proposes that social and organisational support can help reduce the negative effects of stressful experiences by providing employees with practical support, reassurance, and resources to manage difficult situations. Supporting this hypothesis, Schat and Kelloway (2003) examined 225 healthcare employees and found that organisational support moderated {{ic|how strongly? direction?}} the relationships between workplace violence or aggression and outcomes including emotional wellbeing, physical health, and job-related affect. These findings suggest that support from an organisation can help protect employees from some of the harmful outcomes of OV. However, the protective effect was not consistent across all outcomes, as organisational support did not moderate employees' fear of future workplace violence or job neglect. This indicates that support can be valuable in managing the consequences of OV; however, it cannot necessarily remove employees' concerns about further incidents or replace the need to prevent violence itself. ==== Psychological impacts ==== In addition to emotional effects, OV is also associated with increased psychological distress, workplace stress, feelings of helplessness, and reduced psychological safety {{f}}. When violence is ignored or repeatedly occurs, employees may begin to perceive their workplace as unpredictable or unsafe {{f}}. Evidence from research shows that workplace violence is linked to several important psychological outcomes. A systematic review of {{ic|how many studies}} workplace violence against healthcare professionals by Mento and colleagues (2020) reported negative {{ic|how strong?}} cognitive and physical outcomes, including stress and anxiety. These findings suggest that exposure to violence can affect employees’ ability to feel secure and function comfortably at work. Prospective research provides additional evidence that is important when considering longer-term consequences. Pihl-Thingvad et al. (2019) studied the relationship between OV and symptoms of [[wikipedia:Post-traumatic_stress_disorder|post-traumatic stress disorder (PTSD)]] among 1,763 social educators {{where}} over 12 months. Their research examined whether violence was associated with later psychological symptoms and considered the possible indirect roles of time pressure and other work-related stressors. The study found that more frequent exposure to OV was associated with a higher risk of developing PTSD symptoms. The researchers also found that workplace strains, including increased time pressure, burnout, and reduced feelings of safety at work, helped explain the relationship between OV and PTSD. This suggests that the psychological effects of OV are influenced not only by the violent incidents themselves but also by the stressful working conditions that follow or accompany them. ==== Motivation and job satisfaction ==== OV can also affect employees' attitudes towards their work, including motivation, work engagement, job satisfaction, and sense of belonging (Rasool et al., 2021). When employees experience OV, their attention and energy may shift from completing work tasks towards managing fear, avoiding further incidents, or recovering emotionally. For instance, an employee who has been repeatedly intimidated by a supervisor may become reluctant to contribute ideas, ask questions, or participate in team discussions. Similarly, an employee who experiences abuse from customers may begin to dread particular shifts or feel less enthusiastic about responsibilities that they previously enjoyed. Research has identified a relationship between workplace violence and work-related outcomes. Gedik et al. (2023) examined workplace violence, emotional exhaustion, job satisfaction, and turnover intention in a sample of 513 nurses {{where}} during the COVID-19 pandemic. Their findings identified significant relationships among these variables, highlighting the relevance of violence and emotional exhaustion to employees' experiences of job satisfaction and intentions to leave. These findings suggest that OV may contribute to emotional exhaustion, lower job satisfaction, and a greater intention to leave a job. However, further research is needed to examine the relationship between OV and motivation and job satisfaction. This is because research by Gedik et al. (2023) and many other studies, such as Duan et al. (2019) and Liu et al. (2018), focus on nurses and healthcare workers, so their findings may not be equally applicable to every occupation or workplace. Furthermore, the relationship between OV and job satisfaction and motivation does not establish that violence alone explains employees' attitudes. According to a study by Luo et al. (2026), workload, staffing, management quality, job security, and other workplace conditions may influence these relationships. {{RoundBoxTop|theme=9}} ;Key points *Exposure to OV is associated with emotional responses such as fear, anxiety and emotional exhaustion. *Repeated exposure to OV can contribute to long-term psychological effects, including increased destressed symptoms and PTSD. *Social and organisational support can reduce some of the negative consequences. *OV can also influence employees' job satisfaction and workplace engagement. {{RoundBoxBottom}} == Coping with Occupational Violence == Individual coping strategies can help employees manage the emotional and psychological effects of OV. Lazarus and Folkman's (1984) transactional theory of stress and coping distinguished between problem-focused coping, which involves taking action to address a stressful situation, and emotion-focused coping, which aims to manage the emotional responses associated with it. In the context of OV, problem-focused coping could involve seeking assistance from a supervisor, documenting the incidents and preventing further exposure. Emotion-focused coping would include practising meditation and seeking professional counselling. The suitability of each approach depends on the circumstances. Problem-focused coping can be useful when most aspects of the situation can be changed, while emotion-focused coping can help employees manage the emotional impact of incidents they cannot immediately control. [[Motivation and emotion/Book/2019/Social support and emotion|Social support]] is another important coping resource for employees experiencing OV. Support from trusted colleagues, friends, and family members can provide reassurance, reduce feelings of isolation, and give employees an opportunity to discuss their experiences confidentially. Research by Zhao et al. (2015), involving healthcare professionals across 19 hospitals in China, examined the support and prevention strategies employees considered useful following workplace violence. The researchers found that employees exposed to psychological violence frequently reported talking with colleagues, while some also sought comfort from family members. Employees exposed to physical violence reported strategies such as self-defence and talking with colleagues. These findings highlight the importance of interpersonal support following violent incidents. Organisations have a responsibility to respond appropriately to reports of OV provide full support to affected employees. Employees should be able to report incidents to supervisors or management through clear, accessible procedures without fearing blame or retaliation. Appropriate organisational responses may include documenting and investigating incidents, assessing risks, implementing measures to prevent recurrence, following up with affected employees and offering professional resources such as counselling. Overall, coping is not solely an individual responsibility. Employees need supportive workplace systems that take reports seriously and address the circumstances contributing to violence. Eventually, combining individual coping strategies, social support, and effective organisational responses can help employees manage the effects of OV, while prevention further harm. {{f}} {{RoundBoxTop|theme=9}} ;Key points * The transactional theory of stress and coping distinguishes between problem-focused coping, which addresses stressful situation, and emotion-focused coping, which helps employees manage their emotional response. * Social support from friends, family, and trusted colleagues can provide reassurance and help victims of OV manage their feelings and make them feel less isolated. * Employees can benefit from seeking counselling and accessing employee assistance programs where available. * Coping should not be view as an individual responsibility, rather, organisations must provide accessible useful resources for affected employees. {{RoundBoxBottom}} <quiz display="simple"> {Which of the following is an example of Problem-Focused Coping? |type="()"} + Writing down the details and filing an incident report - Practicing deep breathing during a break - Talking to a friend to feel better </quiz> <quiz display="simple"> {True or False: Dealing with workplace violence is solely the individual employee's responsibility? |type="()"} + False - True </quiz> == Preventing and Supporting Employees == [[File:Hands - Art4Good.jpg|thumb|'''Figure 3'''. helping hands ]] Preventing OV requires organisations to establish clear workplace policies and procedures that recognise unacceptable behaviour and outline how incidents will be addressed {{f}}. These policies should explicitly prohibit threats, verbal abuse, bullying, discrimination, and physical aggression, regardless of whether the perpetrator is an employee, supervisor, or customer/client. Organisations should also set clear expectations for professional conduct and communicate the consequences of violating workplace policies, such as disciplinary action. Essentially, employees need confidential and accessible reporting systems that explain how to document incidents, who to contact, and what actions will follow a report. According to the International Labour Organisation (ILO, 2022), prevention requires more than simply informing employees that violence is unacceptable; organisations must establish procedures that ensure harmful behaviour is recognised, reported, investigated, and appropriately addressed. Alongside workplace policies, organisations should implement practical prevention strategies, including regular risk assessments, staff training, and adequate supervision. A systematic review by La Barbiera et al. (2026), which examined eight studies of workplace violence prevention interventions in healthcare settings, found that simulation-based training, structured risk assessment tools, and organisational policies were important components of prevention. Therefore, effective prevention requires organisations to combine employee training with measures that address the workplace conditions contributing to the risks of OV. When an OV incident occurs, organisations must provide immediate support to protect the affected employee from further harm. This may involve separating the employee from the perpetrator, arranging medical attention where necessary, and ensuring the employee has access to a safe environment. Supervisors should respond without dismissing the employee's experience or suggesting that violence is simply an unavoidable part of their job. It is also important to combine immediate emotional support with concrete measures to prevent repeated exposure. {{f}} Longer-term organisational support is also important because the psychological effects of OV could continue even after the initial incident has been resolved. Employers should offer affected employees access to professional counselling or an employee assistance program, alongside appropriate follow-ups to monitor their recovery and overall wellbeing. {{f}} The World Health Organisation (WHO, 2024) recommends responding promptly to violent incidents, providing appropriate assistance, and monitoring workplace violence to evaluate preventive measures. Ultimately, organisational support should extend beyond responding to the initial incident by addressing ongoing psychological needs, reducing the likelihood of further exposure, and helping employees regain their sense of safety at work. {{RoundBoxTop|theme=9}} ;Key points *Effective prevention requires clear workplace policies and appropriate consequences for unacceptable behaviours. *Staff training, regular risk assessment and adequate supervision can help organisations identify and manage situations that increase the risks of OV. *Following an incident, organisations should priorities employees' immediate safety, respond appropriately to reports and take practical actions to prevent repeated exposure. *Long-term support, including counselling, follow-up procedures and workplace adjustments can assist affected employees in managing the psychological consequences of OV. {{RoundBoxBottom}} == Conclusion == Occupational violence is a significant workplace issue that can negatively affect employees' emotional well-being, psychological health, motivation, and job satisfaction. Exposure to OV can result in emotional responses such as fear, anxiety, anger, humiliation, and emotional exhaustion. These experiences may also contribute to longer-term psychological effects, including increased stress, psychological distress, and reduced feelings of safety at work. Although individual coping strategies, including problem-focused coping, emotion-focused coping, and seeking social support can help employees manage these experiences, coping should not be viewed as the sole responsibility of affected employees. Organisations play an essential role in preventing OV and responding appropriately when incidents occur. However, further research is needed to understand how OV affects employees across different occupations; most of the research discussed in this chapter focuses on healthcare workers. Examining other industries, such as hospitality, retail, education, and community services, may provide a broader understanding of how workplace conditions influence employees' experiences of violence and the effectiveness of different prevention and coping strategies. {{RoundBoxTop|theme=7}} ;Take-home message Occupational violence should never be considered a normal part of any job. Whether it is physical, verbal, or psychological, every form of violence deserves to be taken seriously. Remember that seeking help is a sign of strength, and no employee should have to manage these experiences alone. Everyone deserves to feel safe, respected, and supported at work. {{RoundBoxBottom}} == See also == # [[Motivation and emotion/Book/2024/Occupational violence, emotion, and coping for educators|Occupational Violence]] (Book chapter, 2024) # [[wikipedia:Workplace_violence|Workplace Violence]] (Wikipedia) # [[Motivation and emotion/Book/2019/Social support and emotion|Social support and emotion]] (Book chapter, 2019) # [[Motivation and emotion/Book/2011/Social support and achievement|Social support and achievement]] (Book chapter, 2011) == References == {{Hanging indent|1= Ajuwa, M.-E. P., Veyrier, C.-A., Cousin Cabrolier, L., Chassany, O., Marcellin, F., Yaya, I., & Duracinsky, M. (2024). Workplace violence against female healthcare workers: A systematic review and meta-analysis. ''BMJ Open'', ''14''(8), e079396. https://doi.org/10.1136/bmjopen-2023-079396 Bakker, A. B., & Demerouti, E. (2007). The job demands‐resources model: State of the art. ''Journal of Managerial Psychology'', ''22''(3), 309–328. https://doi.org/10.1108/02683940710733115 Brooks, S. K., Dunn, R., Amlôt, R., Rubin, G. J., & Greenberg, N. (2017). Social and occupational factors associated with psychological wellbeing among occupational groups affected by disaster: A systematic review. ''Journal of Mental Health'', ''26''(4), 373–384. https://doi.org/10.1080/09638237.2017.1294732 Cohen, S., & Wills, T. A. (1985). Stress, social support, and the buffering hypothesis. ''Psychological Bulletin'', ''98''(2), 310–357. https://pubmed.ncbi.nlm.nih.gov/3901065/ Duan, X., Ni, X., Shi, L., Zhang, L., Ye, Y., Li, Z., Fan, L., & Wang, Y. (2019). The impact of workplace violence on job satisfaction, job burnout, and turnover intention: The mediating role of social support. Health and Quality of Life Outcomes, 17(1). https://doi.org/10.1186/s12955-019-1164-3 Gedik, Ö., Ülke Şimdi, R., Kıbrıs, Ş., & Sivuk, D. K. (2023). The relationship between workplace violence, emotional exhaustion, job satisfaction and turnover intention among nurses during the COVID-19 pandemic. ''Journal of Research in Nursing : JRN'', ''28''(6–7), 448–466. PubMed. https://doi.org/10.1177/17449871231182837 George, A. S. (2023). Toxicity in the workplace: The silent killer of careers and lives. ''Partners Universal International Innovation Journal'', ''1''(2), 1–21. https://doi.org/10.5281/zenodo.7852087 International Labour Organization. (2022). ''Violence and harassment at work: A practical guide for employers''. International Labour Organization. https://www.ilo.org/publications/violence-and-harassment-work-practical-guide-employers La Barbiera, C., Mento, C., Lombardo, C., Capozza, D., & De Carlo, A. (2026). Success factors in prevention activities and training intervention against workplace violence in healthcare: A systematic review. ''Public Health in Practice'', ''11''(100), 100736. https://doi.org/10.1016/j.puhip.2026.100736 Lanctôt, N., & Guay, S. (2014). The aftermath of workplace violence among healthcare workers: A systematic literature review of the consequences. ''Aggression and Violent Behavior'', ''19''(5), 492–501. https://doi.org/10.1016/j.avb.2014.07.010 Lazarus, R. S., & Folkman, S. (1984). ''Stress, appraisal, and coping''. Springer Publishing Company. Liu, W., Zhao, S., Shi, L., Zhang, Z., Liu, X., Li, L., Li, G., Lou, F., Jia, X., Sun, T., & Ni, X. (2018). Workplace violence, job satisfaction, burnout, perceived organisational support and their effects on turnover intention among Chinese nurses in tertiary hospitals: A cross-sectional study. ''BMJ Open'', ''8''(6), e019525. https://doi.org/10.1136/bmjopen-2017-019525 Luo, H., Li, S., Chen, Y., Tang, X., Tang, H., & Wang, Z. (2026). Relationships between professional value and workplace violence among healthcare workers: An empiricle study from China. ''Frontiers in Medicine'', ''13'', 1768667. PubMed Central. https://doi.org/10.3389/fmed.2026.1768667 Mache, S., Bernburg, M., Würtenberger, A., Groneberg, D. A., & Harth, V. (2026). “We could do more”: Exploring violence prevention measures in German outpatient primary care — a qualitative study. ''BMC Primary Care'', ''27''(1). https://doi.org/10.1186/s12875-026-03509-w Rasool, S. F., Wang, M., Tang, M., Saeed, A., & Iqbal, J. (2021). How toxic workplace environment effects the employee engagement: The mediating role of organizational support and employee wellbeing. ''International Journal of Environmental Research and Public Health'', ''18''(5), 1–17. NCBI. https://doi.org/https://doi.org/10.3390/ijerph18052294 Schat, A. C. H., & Kelloway, E. K. (2003). Reducing the adverse consequences of workplace aggression and violence: The buffering effects of organizational support. ''Journal of Occupational Health Psychology'', ''8''(2), 110–122. https://doi.org/10.1037/1076-8998.8.2.110 World Health Organisation. (2024). ''Violence and harassment''. www.who.int. [https://www.who.int/tools/occupational-&#x20;hazards-in-health-sector/violence-harassment https://www.who.int/tools/occupational-] [https://www.who.int/tools/occupational-&#x20;hazards-in-health-sector/violence-harassment hazards-in-health-sector/violence-harassment] Zhao, S., Liu, H., Ma, H., Jiao, M., Li, Y., Hao, Y., Sun, Y., Gao, L., Hong, S., Kang, Z., Wu, Q., & Qiao, H. (2015). Coping with workplace violence in healthcare settings: Social support and strategies. ''International Journal of Environmental Research and Public Health'', ''12''(11), 14429–14444. https://doi.org/10.3390/ijerph121114429 }} == External links == * [https://www.worksafe.act.gov.au/health-and-safety-portal/safety-topics/mental-health/work-related-violence-or-agression Workplace violence or aggression] (SafeWork ACT) * [https://www.safework.nsw.gov.au/__data/assets/pdf_file/0009/187812/Preventing-and-responding-to-work-related-violence.pdf Preventing and responding to work-related violence] (SafeWork NSW) * [https://www.fwc.gov.au/ Fair Work Commission] (Fair Work Commission Australia) * [https://www.who.int/tools/occupational-hazards-in-health-sector/violence-harassment Violence and harassment] (WHO, 2024) * [https://www.youtube.com/watch?v=PaL1fkkQHlA Occupational Violence and Aggression] (YouTube) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Coping]] [[Category:Motivation and emotion/Book/Emotion]] [[Category:Motivation and emotion/Book/Violence]] [[Category:Motivation and emotion/Book/Work]] krsf65prt6wtsrqrbcwoxagmvs3nhpp 1 323735 2844076 2842888 2026-11-03T02:13:16Z Jtneill 10242 2844076 wikitext text/x-wiki == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:|Hi [[User:{{{1}}}|{{{1}}}]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:53, 23 August 2025 (UTC) |} <!-- Official topic development feedback --> {{METF/2025 |1= <!-- Title --> # Title and sub-title correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> # Promising 2-level heading structure – could benefit from further development and/or refinement # It is unclear why there is an emphasis on the educational context - this isn't part of the title or sub-title - the topic is broader, although education is one context in which OV risk is heightened # The structure is overly detailed e.g., need to be more selective about focusing on the top theories, not lots of theories <!-- Alignment with focus questions --> # Insufficient alignment between sub-title, focus questions, and top-level headings |3= <!-- Overview--> # Basic <!-- Scenario --> # A scenario or case study is planned presented in a feature box with an image at the start of this section # Add an image to the scenario to help attract reader interest <!-- Description --> # An insufficient description is provided # The topic is broader than OV in a school context (e.g., also consider other at-risk areas such as healthcare) # This topic is about what psychological science has learned across contexts rather than in one particular context # Find and use the best psychological theory and research about this problem <!-- Focus questions --> # Focus questions lack sufficient focus on "What are the emotional impacts of occupational violence and how can employees cope?" # Develop closer alignment between the sub-title, focus questions, and top-level headings |4= <!-- Key points--> <!-- Overall --> # Promising, but insufficient development of key points that address the sub-title question # Overly focused on theories and schools # Insufficiently focused on synthesising the best psychological theory and research about cross-sector (a) emotional impacts of OV; and (b) how employees can cope <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples <!-- Citations --> # Basic use of citations <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings <!-- Conclusion --> # Conclusion (the most important section) hasn't been developed # What might the take-home, practical messages be? (What are the answer(s) to the question(s) in the sub-title and/or focus questions?) |5= <!-- Figure --> # A figure(s) are presented and captioned # Relevance of figure is unclear <!-- Caption --> # Figure caption(s) could better explain how the image connects to key points being made in the main text #<!-- Cite --> # Figure(s) are cited at least once in the main text <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Basic # The references are overly focused on OV in schools and insufficiently focused on cross-sectorial emotional impacts and coping strategies <!-- Systematic reviews --> # At least one relevant systematic review and/or meta-analysis has been identified <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]]) ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Basic ## These links are overly focused on (a) Australia; (b) education ## Only include links directly related to the sub-title ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Used effectively <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 02]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:53, 23 August 2025 (UTC) == Exemplar Template == Great progress so far. You can also refer to the [[Template:Motivation and emotion/Book chapter structure|template]] example provided to get a better idea of how you can lay out the page or to make sure you include everything. --[[User:Diaz Chas|Diaz Chas]] ([[User talk:Diaz Chas|discuss]] • [[Special:Contributions/Diaz Chas|contribs]]) 10:13, 26 August 2026 (UTC) [[User:Diaz Chas|Diaz Chas]] ([[User talk:Diaz Chas|discuss]] • [[Special:Contributions/Diaz Chas|contribs]]) 10:11, 26 August 2026 (UTC)[[User:Diaz Chas|Diaz Chas]] ([[User talk:Diaz Chas|discuss]] • [[Special:Contributions/Diaz Chas|contribs]]) 10:08, 26 August 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # It appears that some sub-headings are planned but not yet formatted using wiki style headings (see Tutorial 2) <!-- Other ---> # Inconsistent use of OV abbreviation <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure |3= <!-- Overview--> # Scenario, image, evocative description of the problem/topic, and focus questions <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section <!-- Description --> # A basic description of the problem/topic is planned or presented # Introduce topic using plain English <!-- Style --> # Use present, rather than future, tense (fixed) <!-- Focus questions --> # Focus questions are promising, but can be improved to be more singular, correct grammar, and bullet points (as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |4= <!-- Key points--> <!-- Overall --> # Basic development # Provide more detailed edit summaries <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Insufficient use of citations <!-- Conclusion --> # Conclusion is underdeveloped (vague) # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Relevant figure(s) are presented and captioned <!-- Caption --> # Figure caption(s) should include '''Figure X'''. ... (fixed) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Very good <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Citations --> # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## author initials |8= <!-- Resources --> <!-- See also --> # See also ## Very good ## Use alphabetical order <!-- External links --> # External links ## Move academic sources into the "References" sections and provide in-text citation ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Basic but effective <!-- Description about self --> # Brief description about self – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:06, 5 September 2026 (UTC) == General feedback and edits == Hey! I think your article reads well and has a good flow. I think Emma's scenario makes the topic really relatable, and the discussion of risks of OV provide a good focus. A few suggestions # returning to the example of Emma throughout your chapter could serve to apply your discussions to a specific example. # I think there are a few missing reference-list entries. e.g. Bakker and Demrouti (2007) is in the text but not in the reference list. # I would emphasise the take away points in your conclusion. There are so many potential take-aways from your article that could be applied to almost everyone! # I would recommend including (and referencing) some more images or learning features in your chapter to break up the text so it is a bit more readable. If you want to discuss, reach out at U3216851@uni.canberra.edu.au Thanks! --[[User:U3216851|U3216851]] ([[User talk:U3216851|discuss]] • [[Special:Contributions/U3216851|contribs]]) 03:28, 10 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # The main area(s) for potential improvement: #* integrate more strongly with other chapters by using embedded links to related topics <!-- Overall - Edit summaries --> #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit #* Increase use of academic, peer-reviewed citations to support claims <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FOccupational_violence%2C_emotion%2C_and_coping&diff=2842885&oldid=2839198 these copyedits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied <!-- Theory – Builds on --> # Builds effectively on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Reasonably good use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # More detail about key studies would be ideal # What is the prevalence of OV? # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. <!-- Research – Critical thinking --> # Good [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Very good integration between the most relevant theory(ies) and the best research # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised <!-- Conclusion – Focus questions --> # Take-away messages for each focus question could be spelt out more clearly <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is very good <!-- Written expression – Layout --> # Layout ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## See earlier comment about [[#Heading casing|heading casing]] ## The Overview and Conclusion should not have subheadings (fixed) <!-- Written expression – Grammar --> # Grammar and spelling are excellent <!-- Written expression – Abbreviations --> ## Abbreviations ### Once an abbreviation has been established (e.g., OV), use it consistently afterwards <!-- Written expression – Copy editing --> # Copy editing ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] <!-- Written expression – Figures --> ## Figures ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. <!-- Written expression – Figures - Citations --> ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – Citations --> # In many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses <!-- Written expression – References --> ## References use excellent APA style <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent use of learning features <!-- Learning features – Wikipedia embedded links --> # Very good use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Basic use of figure(s) # No use of table(s) # Excellent use of feature box(es) # Basic use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Excellent use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing ## Use alphabetical order <!-- Learning features – External links --> # Basic use of the [[{{PAGENAME}}#External links|External links]] section ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Target an international audience |9= <!-- Social contribution comments... --> # ~6 logged, useful, mostly minor to moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~1 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # The Wiki Commons upload was removed (likely due to copyright issues) }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 21:53, 27 October 2026 (UTC) hdd0tmgdn7dfe7416et9u3ys7d9yqwv 0 324196 2844028 2805483 2026-11-02T21:29:31Z Francesco Cattafi 2920091 Minor corrections (mostly in the references), following the reimplementation in the original wikipedia article 2844028 wikitext text/x-wiki {{Article info | last1 = Cattafi | orcid1 = 0000-0001-9785-0770 | first1 = Francesco | affiliation1 = Julius-Maximilians-Universität Würzburg | correspondence1 = francesco.cattafi91@gmail.com | last2 = Miyamoto | orcid2 = 0000-0003-1780-2539 | first2 = David | affiliation2 = Queen's University at Kingston | correspondence2 = ian.david.miyamoto@gmail.com | journal = WikiJournal of Science | et_al = true | w1 = Diffeology | from w1 = true | keywords = Diffeology, Diffeological space, Smooth manifold, Smooth space, Singular space | license = CC-BY-SA 4.0 | submitted = 2025-09-21 | abstract = In [[w:mathematics|mathematics]], a '''diffeology''' on a set generalizes the concept of a smooth atlas of a [[w:differentiable manifold|differentiable manifold]], by declaring only what constitutes the "smooth parametrizations" into the set. A diffeological space is a set equipped with a diffeology. Many of the standard tools of [[w:differential geometry|differential geometry]] extend to diffeological spaces, which beyond manifolds include arbitrary quotients of manifolds, arbitrary subsets of manifolds, and spaces of mappings between manifolds. }} == Introduction == === Calculus on "smooth spaces" === The [[w:differential calculus|differential calculus]] on <math>\mathbb{R}^n</math>, or, more generally, on real finite dimensional [[w:vector space|vector space]]s, is one of the most impactful successes of modern mathematics. Fundamental to its basic definitions and theorems is the linear structure of the underlying space.<ref name="Spiv65"/><ref name="Mun91"/> The field of [[w:differential geometry|differential geometry]] establishes and studies the extension of the classical differential calculus to non-linear spaces. This extension is made possible by the definition of a [[w:differentiable manifold|smooth manifold]], which is also the starting point for diffeological spaces. A smooth <math>n</math>-dimensional manifold (hereafter "manifold") is a set <math>M</math> equipped with a maximal [[w:smooth atlas|smooth atlas]]. This data is uniquely determined by a collection of injective functions, called '''local parametrizations''', of the form <math>\phi\colon U \to M</math>, where <math>U</math> is an open subset of <math>\mathbb{R}^n</math>, satisfying some mutual-compatibility relations (the inverse maps <math>\phi^{-1}\colon \phi(U) \to U</math> are called [[w:Chart (mathematics)|charts]]). The local parametrizations of a manifold perform two distinct functions, which are often syncretized:<ref name="KobNom96"/><ref name="Tu11"/><ref name="Lee13"/> * They dictate the local structure of the manifold. The local parametrization <math>\phi\colon U \to M</math> identifies its image in <math>M</math> with its domain <math>U</math>. This is convenient because the latter is simply an open subset of a [[w:Euclidean space|Euclidean space]]. * They define the class of smooth maps between manifolds. These are the maps to which the differential calculus extends. In particular, the local parametrizations determine smooth functions (smooth maps <math>M \to \mathbb{R}</math>), smooth [[w:curve|curve]]s (smooth maps <math>[0,1] \to M</math>), smooth [[w:Homotopy|homotopies]] between curves (smooth maps <math>[0,1]^2 \to M</math>), etc. A diffeology generalizes the structure of a smooth manifold by abandoning the first requirement for an atlas, namely that the local parametrizations give a local model of the space, while retaining the ability to discuss smooth maps into the space.<ref name="Igl13"/><ref name="Igl21" /><ref name="Igl22" /> === Informal definition === A '''diffeological space''' is a set <math>X</math> equipped with a '''diffeology''': a collection of maps<math display="block">\{p\colon U \to X\mid U \text{ is an open subset of }\mathbb{R}^n, \text{ and } n \geq 0\},</math>whose members are called '''plots''', that satisfies some axioms. The plots are not required to be injective, and can (indeed, must) have as domains the open subsets of arbitrary Euclidean spaces. A smooth manifold can be viewed as a diffeological space which is locally diffeomorphic to <math>\mathbb{R}^n</math>. In general, while not giving local models for the space, the axioms of a diffeology still ensure that the plots induce a coherent notion of smooth functions, smooth curves, smooth homotopies, etc. Diffeology is therefore suitable to treat objects more general than manifolds.<ref name="Igl13"/><ref name="Igl21"/><ref name="Igl22"/> === Motivation === The primary motivation to generalize manifolds to diffeological spaces is that many spaces that arise in the course of study of manifolds are not themselves manifolds, but nevertheless carry canonical diffeologies; thus diffeology supplies geometric tools to handle these non-manifolds. Such spaces often arise in three ways: taking a subset of a manifold, taking a quotient of a manifold, and taking a collection of smooth maps between two manifolds. An arbitrary subset of a manifold is not necessarily a [[w:Submanifold|submanifold]]. In categorical terms, this means the [[w:Category_(mathematics)|category]] of manifolds is not [[w:Complete_category|complete]]. The subset can fail to be a submanifold even when it is captured by relatively simple functions. For instance, the [[w:Semicubical_parabola|semi-cubic]] <math>\{(x,y) \in \mathbb{R}^2 \mid x^2=y^3\}</math> is not an embedded submanifold of the plane <math>\mathbb{R}^2</math>, due to the [[w:Cusp_(singularity)|cusp point]] at the origin. On the other hand, the category of diffeological spaces is complete, meaning that every subset of a diffeological space inherits a natural diffeology. The semi-cubic, as a diffeological space, is isomorphic to <math>\mathbb{R}</math>.<ref name="KarMiyWat24" /> An arbitrary quotient of a manifold is not necessarily a manifold. In categorical terms, this means the category of manifolds is not [[w:Complete_category|co-complete]]. An example is the irrational torus <math>T_\alpha := \mathbb{R}/(\mathbb{Z} + \alpha \mathbb{Z})</math>, where <math>\alpha</math> is a fixed irrational number. These irrational tori appear naturally: e.g. in [[w:Geometric_quantization|geometric quantization]] they arise as the period groups of [[w:Symplectic_manifold|symplectic forms]] with non-integral [[w:Cohomology|cohomology]]. An irrational torus is not a manifold, because it carries the trivial topology. On the other hand, the category of diffeological spaces is co-complete, meaning that every quotient of a diffeological space inherits a natural diffeology. Two irrational tori, <math>T_\alpha</math> and <math>T_\beta</math>, are isomorphic as diffeological spaces if and only if <math>\alpha</math> and <math>\beta</math> are related by a [[w:Homography#Homographies_of_a_projective_line|homography]] with integer coefficients.<ref name="DonIgl85" /> Finally, the space of smooth maps <math>C^\infty(M,N)</math> between manifolds <math>M</math> and <math>N</math> is not a (finite-dimensional) manifold when <math>M</math> and <math>N</math> have positive dimension. In categorical terms, this means the category of manifolds is not [[w:Cartesian_closed_category|Cartesian closed]]. On the other hand, the category of diffeological spaces is Cartesian closed, meaning that every [[w:Mapping_space|mapping space]] between diffeological spaces carries a natural diffeology. For <math>C^\infty(M,N)</math>, this diffeology is the collection of maps <math display="block">\{p\colon U \to \mathcal{C}^\infty(M,N) \mid \text{ the map }U \times M \to N, \ (r,x) \mapsto p(r)(x) \text{ is smooth}\}.</math> With this structure, the diffeologically smooth curves from <math>[0,1]</math> into <math>C^\infty(M,N)</math> (a notion which is now rigorously defined) can be shown to correspond precisely to the smooth homotopies between smooth maps <math>M \to N</math>.<ref name="Igl13" /><ref name="Igl21" /><ref name="Igl22" /> === History === The concept of diffeology was first introduced by [[w:Jean-Marie Souriau|Jean-Marie Souriau]] in the 1980s under the name ''espace différentiel.''<ref name="Sour80"/><ref name="Sour84"/> Souriau's motivating application for diffeology was to uniformly handle the infinite-dimensional groups arising from his work in [[w:geometric quantization|geometric quantization]]. Thus the notion of diffeological group preceded the more general concept of a diffeological space. Souriau's diffeological program was taken up by his students, particularly [[w:Paul G. Donato|Paul Donato]]<ref name="Don84"/> and [[w:Patrick Iglesias-Zemmour|Patrick Iglesias-Zemmour]],<ref name="Igl85"/> who completed early pioneering work in the field. A structure similar to diffeology was introduced by [[w:Kuo-Tsaï Chen|Kuo-Tsaï Chen]] (陳國才, ''Chen Guocai'') in the 1970s, in order to formalize certain computations with path integrals. Chen's definition used [[w:convex set|convex set]]s instead of open sets for the domains of the plots.<ref name="Chen77"/> The similarity between diffeological and "Chen" structures can be made precise by viewing both as concrete sheaves over the appropriate concrete site.<ref name="BaezHof11"/> == Formal definition == A '''diffeology''' on a set ''<math>X</math>'' consists of a collection of maps, called '''plots''', from [[w:Open set|open subsets]] of <math>\mathbb{R}^n</math> (for all ''<math>n \geq 0</math>'') to ''<math>X</math>'' such that the following axioms hold: * '''Covering axiom''': every constant map is a plot. * '''Locality axiom''': for a given map ''<math>p\colon U \to X</math>'', if every point in ''<math>U</math>'' has a [[w:Neighborhood (topology)|neighborhood]] ''<math>V \subset U</math>'' such that ''<math>p|_V</math>'' is a plot, then ''<math>p</math>'' itself is a plot. * '''Smooth compatibility axiom''': if ''<math>p</math>'' is a plot, and ''<math>F</math>'' is a [[w:smooth map|smooth map]] from an open subset of some <math>\mathbb{R}^m</math> into the domain of ''<math>p</math>'', then the composite ''<math>p \circ F</math>'' is a plot. Note that the domains of different plots can be subsets of <math>\mathbb{R}^n</math> for different values of ''<math>n</math>''; in particular, any diffeology contains the constant maps ''<math>x: \mathbb{R}^0 \to X</math>'', for all elements ''<math>x</math>'' of the underlying set ''<math>X</math>'', as the plots with ''<math>n = 0</math>''. A set together with a diffeology is called a '''diffeological space'''. More abstractly, a diffeological space is a concrete [[w:Sheaf (mathematics)|sheaf]] on the [[w:Site (mathematics)|site]] of open subsets of <math>\mathbb{R}^n</math>, for all ''<math>n \geq 0</math>'', and [[w:open cover|open cover]]s.<ref name="BaezHof11"/> === Morphisms === A map between diffeological spaces is called '''smooth''' if and only if its composite with any plot of the first space is a plot of the second space. It is called a '''diffeomorphism''' if it is smooth, [[w:bijective|bijective]], and its [[w:Inverse function|inverse]] is also smooth. Equipping the open subsets of Euclidean spaces with their standard diffeology (as defined in the next section), the plots into a diffeological space ''<math>X</math>'' are precisely the smooth maps from ''<math>U</math>'' to ''<math>X</math>''. Diffeological spaces constitute the objects of a category, denoted by <math>\mathsf{Dflg}</math>, whose [[w:morphism|morphism]]s are smooth maps. The category <math>\mathsf{Dflg}</math> is closed under many categorical operations: for instance, it is Cartesian closed, complete and cocomplete, and more generally it is a [[w:quasitopos|quasitopos]].<ref name="BaezHof11"/> === D-topology === Any diffeological space is a [[w:topological space|topological space]] when equipped with the '''D-topology''':<ref name="Igl85" /> the [[w:final topology|final topology]] such that all plots are [[w:Continuous function (topology)|continuous]] (with respect to the [[w:Euclidean topology|Euclidean topology]] on <math>\mathbb{R}^n</math>). In other words, a subset <math>U \subset X</math> is open if and only if <math>p^{-1}(U)</math> is open for any plot <math>p</math> on <math>X</math>. Actually, the D-topology is completely determined by smooth curves, i.e., a subset <math>U \subset X</math> is open if and only if <math>c^{-1}(U)</math> is open for any smooth map <math>c\colon \mathbb{R} \to X</math>.<ref name="ChrSinWu14"/> The D-topology is automatically [[w:Locally path connected|locally path-connected]]<ref name="Laub06"/> A smooth map between diffeological spaces is automatically continuous between their D-topologies.<ref name="Igl13" /> Therefore we have the functor <math>D\colon \mathsf{Dflg} \to \mathsf{Top}</math>, from the category of diffeological spaces to the category of topological spaces, which assigns to a diffeological space its D-topology. This functor realizes <math>\mathsf{Dflg}</math> as a [[w:concrete category|concrete category]] over <math>\mathsf{Top}</math>. === Additional structures === A Cartan-De Rham calculus can be developed in the framework of diffeologies, as well as a suitable adaptation of the notions of [[w:fiber bundle|fiber bundle]]s, [[w:homotopy|homotopy]], etc.<ref name="Igl13"/> There are also several non-equivalent definitions of [[w:tangent space|tangent space]]s and [[w:tangent bundle|tangent bundle]]s for diffeological spaces.<ref name="ChrWu14"/><ref name="Bloh24"/><ref name="Tah26"/> == Examples == === First examples === Any set with at least two elements carries at least two (different) diffeologies: * the '''coarse''' (or trivial, or indiscrete) diffeology, consisting of every map into the set. This is the largest possible diffeology. The corresponding D-topology is the [[w:trivial topology|trivial topology]]. * the '''discrete''' (or fine) diffeology, consisting of the locally constant maps into the set. This is the smallest possible diffeology. The corresponding D-topology is the [[w:discrete topology|discrete topology]]. The empty set and any singleton admit a unique diffeology (the two diffeologies above coincide). Any topological space can be endowed with the '''continuous''' diffeology, whose plots are the [[w:Continuous function (topology)|continuous]] maps. The Euclidean space <math>\mathbb{R}^n</math> admits several diffeologies beyond those listed above. * The '''standard''' diffeology on <math>\mathbb{R}^n</math> consists of those maps <math>p:U \to \mathbb{R}^n</math> which are smooth in the usual sense of multivariable calculus. * The '''wire''' (or spaghetti) diffeology on <math>\mathbb{R}^n</math> is the diffeology whose plots factor locally through <math>\mathbb{R}</math>. Denote <math>\mathbb{R}^n</math> equipped with the wire diffeology by <math>X</math>. A map <math>p\colon U \to X</math> is a plot if and only if for every <math>u \in U</math> there is an open neighbourhood <math>V \subseteq U</math> of <math>u</math> such that <math>p|_V = q \circ F</math> for two smooth functions <math>F\colon V \to \mathbb{R}</math> and <math>q\colon \mathbb{R} \to \mathbb{R}^n</math>. This diffeology does not coincide with the standard diffeology on <math>\mathbb{R}^n</math> when <math>n\geq 2</math>: for instance, the identity <math display="inline">\mathbb{R}^n \to X</math> is not a plot for the wire diffeology. Equivalently, the identity, viewed as a map from <math>\mathbb{R}^n</math> with its standard structure to <math>X</math>, is not smooth.<ref name="Igl13" /> * The previous example can be enlarged to diffeologies whose plots factor locally through <math>\mathbb{R}^r</math>, yielding the '''rank-<math>r</math>-restricted''' diffeology on a smooth manifold <math>M</math>: a map <math>U \to M</math> is a plot if and only if it is smooth and the rank of its [[w:Pushforward (differential)|differential]] is less than or equal than <math>r</math>. For <math>r=1</math> one recovers the wire diffeology.<ref name="Bloh24" /> === Relation to other smooth spaces === Diffeological spaces generalize manifolds, but they are far from the only mathematical objects to do so. For instance manifolds with corners, orbifolds, and Fréchet manifolds are all well-established alternatives. This subsection makes precise the extent to which these spaces are diffeological. We view <math>\mathsf{Dflg}</math> as a concrete category over the category of topological spaces <math>\mathsf{Top}</math> via the D-topology functor <math>D\colon\mathsf{Dflg} \to \mathsf{Top}</math>. If <math>U\colon \mathsf{C} \to \mathsf{Top}</math> is another concrete category over <math>\mathsf{Top}</math>, we say that a functor <math>E\colon \mathsf{C} \to \mathsf{Dflg}</math> is an embedding (of concrete categories) if it is injective on objects and faithful, and <math>D \circ E = U</math>. To specify an embedding, we need only describe it on objects; it is necessarily the identity map on arrows. We will say that a diffeological space <math>X</math> is '''locally modeled''' by a collection of diffeological spaces <math>\mathcal{E}</math> if around every point <math>x \in X</math>, there is a D-open neighbourhood <math>U</math>, a D-open subset <math>V</math> of some <math>E \in \mathcal{E}</math>, and a diffeological diffeomorphism <math>U \to V</math>.<ref name="Igl13"/><ref name="Nest21"/> ==== Manifolds ==== The category of finite-dimensional smooth manifolds (allowing those with connected components of different dimensions) fully embeds into <math>\mathsf{Dflg}</math>. The embedding <math>y</math> assigns to a smooth manifold <math>M</math> the canonical diffeology<math display="block">\{p\colon U \to M \mid p \text{ is smooth in the usual sense}\}.</math>In particular, a diffeologically smooth map between manifolds is smooth in the usual sense, and the D-topology of <math>y(M)</math> is the original topology of <math>M</math>. The [[w:Image (category theory)#Essential Image|essential image]] of this embedding consists of those diffeological spaces that are locally modeled by the collection <math>\{y(\mathbb{R}^n)\}</math>, and whose D-topology is [[w:Hausdorff space|Hausdorff]] and [[w:Second-countable space|second-countable]].<ref name="Igl13"/> ==== Manifolds with boundary or corners ==== The category of finite-dimensional smooth [[w:Manifold#Manifold with boundary|manifolds with boundary]] (allowing those with connected components of different dimensions) similarly fully embeds into <math>\mathsf{Dflg}</math>. The embedding is defined identically to the smooth case, except "smooth in the usual sense" refers to the standard definition of smooth maps between manifolds with boundary. The essential image of this embedding consists of those diffeological spaces that are locally modeled by the collection <math>\{y(O) \mid O \text{ is a half-space}\}</math>, and whose D-topology is Hausdorff and second-countable. The same can be done in more generality for [[w:Manifold with corners|manifolds with corners]], using the collection <math>\{y(O) \mid O \text{ is an orthant}\}</math>.<ref name="GurIgl19"/> ==== Fréchet and Banach manifolds ==== The category of [[w:Fréchet manifold|Fréchet manifold]]s similarly fully embeds into <math>\mathsf{Dflg}</math>. Once again, the embedding is defined identically to the smooth case, except "smooth in the usual sense" refers to the standard definition of smooth maps between Fréchet spaces. The essential image of this embedding consists of those diffeological spaces that are locally modeled by the collection <math>\{y(E) \mid E \text{ is a Fréchet space}\}</math>, and whose D-topology is Hausdorff. The embedding restricts to one of the category of [[w:Banach manifold|Banach manifold]]s. Historically, the case of Banach manifolds was proved first, by Hain,<ref name="Hain79"/> and the case of Fréchet manifolds was treated later, by Losik.<ref name="Los92"/><ref name="Los94"/> The category of manifolds modeled on [[w:convenient vector space|convenient vector space]]s also similarly embeds into <math>\mathsf{Dflg}</math>.<ref name="FrolKrieg88"/><ref name="Miy25"/> ==== Orbifolds ==== A (classical) [[w:orbifold|orbifold]] <math>X</math> is a space that is locally modeled by quotients of the form <math>\mathbb{R}^n/\Gamma</math>, where <math>\Gamma</math> is a [[w:Finite group|finite subgroup]] of linear transformations. On the other hand, each model <math>\mathbb{R}^n/\Gamma</math> is naturally a diffeological space (with the quotient diffeology discussed below), and therefore the (inverse of the) orbifold charts generate a diffeology on <math>X</math>. This diffeology is uniquely determined by the orbifold structure of <math>X</math>. Conversely, a diffeological space that is locally modeled by the collection <math>\{\mathbb{R}^n/\Gamma\}</math> (and with Hausdorff D-topology) carries a classical orbifold structure that induces the original diffeology, wherein the local diffeomorphisms are the (the inverses of the) orbifold charts. Such a space is called a diffeological orbifold.<ref name="IglKarZad10"/> Whereas diffeological orbifolds automatically have a notion of smooth map between them (namely diffeologically smooth maps in <math>\mathsf{Dflg}</math>), the notion of a smooth map between classical orbifolds is not standardized. If orbifolds are viewed as [[w:differentiable stack|differentiable stack]]s presented by étale proper [[w:Lie groupoid|Lie groupoid]]s, then there is a functor from the underlying 1-category of orbifolds, and equivalent maps-of-stacks between them, to <math>\mathsf{Dflg}</math>. Its essential image consists of diffeological orbifolds, but the functor is neither faithful nor full.<ref name="Wat17" /><ref name="Miy24" /> == Constructions == === Intersections === If a set ''<math>X</math>'' is given two different diffeologies, their [[w:intersection|intersection]] is a diffeology on ''<math>X</math>'', called the '''intersection diffeology''', which is finer than both starting diffeologies. The D-topology of the intersection diffeology is finer than the intersection of the D-topologies of the original diffeologies. === Products === If ''<math>X</math>'' and ''<math>Y</math>'' are diffeological spaces, then the '''product''' diffeology on the [[w:Cartesian product|Cartesian product]] ''<math>X \times Y</math>'' is the diffeology generated by all products of plots of ''<math>X</math>'' and of ''<math>Y</math>''. Precisely, a map <math>p\colon U \to X \times Y</math> necessarily has the form <math>p(u) = (x(u),y(u))</math> for maps <math>x\colon U \to X</math> and <math>y\colon U \to Y</math>. The map <math>p</math> is a plot in the product diffeology if and only if <math>x</math> and <math>y</math> are plots of <math>X</math> and <math>Y</math>, respectively. This generalizes to products of arbitrary collections of spaces. The D-topology of ''<math>X \times Y</math>'' is the coarsest delta-generated topology containing the [[w:product topology|product topology]] of the D-topologies of ''<math>X</math>'' and ''<math>Y</math>''; it is equal to the product topology when ''<math>X</math>'' or ''<math>Y</math>'' is [[w:locally compact|locally compact]], but may be finer in general.<ref name="ChrSinWu14"/> ===Coproducts=== If <math>X</math> and <math>Y</math> are diffeological spaces, then the '''coproduct''' diffeology on the [[w:Disjoint_union|disjoint union]] <math> X \amalg Y</math> is the diffeology consisting of all parametrizations that are locally a plot of <math>X</math> or of <math>Y</math>. Precisely, a map <math>p\colon U \to X \amalg Y</math> is a plot in the coproduct diffeology if and only if around every point in <math>U</math>, there is a neighbourhood on which <math>p</math> restricts to a plot of either <math>X</math> or of <math>Y</math>. The D-topology of <math>X \amalg Y</math> is the [[w:Disjoint_union_(topology)|disjoint union]] of the D-topologies of <math>X</math> and <math>Y</math>. === Pullbacks === Given a map ''<math>f\colon X \to Y</math>'' from a set <math>X</math> to a diffeological space <math>Y</math>, the '''pullback''' diffeology on ''<math>X</math>'' consists of those maps ''<math>p\colon U \to X</math>'' such that the composition ''<math>f \circ p</math>'' is a plot of ''<math>Y</math>''. In other words, the pullback diffeology is the smallest diffeology on ''<math>X</math>'' making ''<math>f</math>'' smooth. If ''<math>X</math>'' is a [[w:subset|subset]] of the diffeological space ''<math>Y</math>'', then the '''subspace''' diffeology on ''<math>X</math>'' is the pullback diffeology induced by the inclusion <math>X \hookrightarrow Y</math>. In this case, the D-topology of ''<math>X</math>'' is equal to the [[w:subspace topology|subspace topology]] of the D-topology of ''<math>Y</math>'' if ''<math>Y</math>'' is open, but may be finer in general. === Pushforwards === Given a map ''<math>f\colon X \to Y</math>'' from diffeological space ''<math>X</math>'' to a set <math>Y</math>, the '''pushforward''' diffeology on ''<math>Y</math>'' is the diffeology generated by the compositions ''<math>f \circ p</math>'', for plots ''<math>p\colon U \to X</math>'' of ''<math>X</math>''. In other words, the pushforward diffeology is the smallest diffeology on ''<math>Y</math>'' making ''<math>f</math>'' smooth. If ''<math>X</math>'' is a diffeological space and ''<math>\sim</math>'' is an [[w:equivalence relation|equivalence relation]] on ''<math>X</math>'', then the '''quotient''' diffeology on the [[w:quotient set|quotient set]] ''<math>X/{\sim}</math>'' is the pushforward diffeology induced by the quotient map <math>X \to X/{\sim}</math>. The D-topology on ''<math>X/{\sim}</math>'' is the [[w:quotient topology|quotient topology]] of the D-topology of ''<math>X</math>''. Note that this topology may be trivial without the diffeology being trivial. Quotients often give rise to non-manifold diffeologies. For example, the set of [[w:real number|real number]]s '''<math>\mathbb{R}</math>''' is a smooth manifold. The quotient <math>\mathbb{R}/(\mathbb{Z} + \alpha \mathbb{Z})</math>, for some [[w:Irrational number|irrational]] ''<math>\alpha</math>'', called the '''irrational torus''', is a diffeological space diffeomorphic to the quotient of the regular [[w:Torus|2-torus]] <math>\mathbb{R}^2/\mathbb{Z}^2</math> by a line of [[w:slope|slope]] ''<math>\alpha</math>''. It has a non-trivial diffeology, although its D-topology is the [[w:trivial topology|trivial topology]].<ref name="DonIgl85"/> === Functional diffeologies === The '''functional''' diffeology on the set <math>\mathcal{C}^{\infty}(X,Y)</math> of smooth maps between two diffeological spaces <math>X</math> and <math>Y</math> is the diffeology whose plots are the maps <math>p\colon U \to \mathcal{C}^{\infty}(X,Y)</math> such that<math display="block">U \times X \to Y, \quad (u,x) \mapsto p(u)(x)</math>is smooth with respect to the product diffeology of <math>U \times X</math>. When ''<math>X</math>'' and ''<math>Y</math>'' are manifolds, the D-topology of <math>\mathcal{C}^{\infty}(X,Y)</math> is the smallest [[w:Locally path connected|locally path-connected]] topology containing the [[w:Whitney Topologies|Whitney <math>C^\infty</math> topology]].<ref name="ChrSinWu14"/> Taking the subspace diffeology of a functional diffeology, one can define diffeologies on the space of [[w:Section (fiber bundle)|sections]] of a [[w:Fiber bundle|fibre bundle]], or the space of bisections of a [[w:Lie groupoid|Lie groupoid]], etc. If <math>M</math> is a compact smooth manifold, and <math>F \to M</math> is a smooth fiber bundle over <math>M</math>, then the space of smooth sections <math>\Gamma(F)</math> of the bundle is frequently equipped with the structure of a Fréchet manifold.<ref name="Ham82"/> Upon embedding this Fréchet manifold into the category of diffeological spaces, the resulting diffeology coincides with the subspace diffeology that <math>\Gamma(F)</math> inherits from the functional diffeology on <math>\mathcal{C}^\infty(M,F)</math>.<ref name="Wal12"/> == Distinguished maps == Analogous to the notions of [[w:Submersion (mathematics)|submersions]] and [[w:Immersion (mathematics)|immersions]] between manifolds, there are two special classes of morphisms between diffeological spaces. A '''subduction''' is a surjective function ''<math>f\colon X \to Y</math>'' between diffeological spaces such that the diffeology of ''<math>Y</math>'' is the pushforward of the diffeology of ''<math>X</math>''. Similarly, an '''induction''' is an injective function ''<math>f\colon X \to Y</math>'' between diffeological spaces such that the diffeology of ''<math>X</math> ''is the pullback of the diffeology of ''<math>Y</math>''. Subductions and inductions are automatically smooth. It is instructive to consider the case where ''<math>X</math>'' and ''<math>Y</math>'' are smooth manifolds. * Every surjective [[w:Submersion (mathematics)|submersion]] <math>f\colon X \to Y</math> is a subduction. * A subduction need not be a surjective submersion. One example is <math display="block">f\colon \mathbb{R}^2 \to \mathbb{R}, \quad f(x,y) := xy.</math> * An injective [[w:Immersion (mathematics)|immersion]] need not be an induction. One example is the parametrization of the "figure-eight,"<math display="block">f\colon \left(-\frac{\pi}{2}, \frac{3\pi}{2}\right) \to \mathbb{R^2}, \quad f(t) := (2\cos(t), \sin(2t)).</math> * An induction need not be an injective immersion. One example is the "semi-cubic,"<ref name="KarMiyWat24"/><ref name="Jor82"/><math display="block">f\colon \mathbb{R} \to \mathbb{R}^2, \quad f(t) := (t^2, t^3).</math> In the category of diffeological spaces, subductions are precisely the strong [[w:epimorphism|epimorphism]]s, and inductions are precisely the strong [[w:monomorphism|monomorphism]]s.<ref name="Bloh24"/> A map that is both a subduction and induction is a diffeomorphism. == References == {{Reflist|refs= <ref name="Spiv65">{{Cite book |last=Spivak |first=Michael |author-link=w:Michael Spivak |title=[[w:Calculus on Manifolds (book)|Calculus on manifolds: a modern approach to classical theorems of advanced calculus]] |date=1965 |publisher=[[w:Benjamin Cummings|Benjamin Cummings]] |isbn=978-0-8053-9021-6 |series=Mathematics monograph series |location=Redwood city (Calif.)}}</ref> <ref name="Mun91">{{Cite book |last=Munkres |first=James R. |author-link=w:James Munkres |url=https://www.taylorfrancis.com/books/9780429962691 |title=Analysis on Manifolds |date=2018-02-19 |publisher=[[w:CRC Press|CRC Press]] |isbn=978-0-429-49414-7 |edition=1 |doi=10.1201/9780429494147}}</ref> <ref name="KobNom96">{{Cite book |last1=Kobayashi |first1=Shōshichi |author-link1=w:Shoshichi Kobayashi |title=[[w:Foundations of Differential Geometry|Foundations of differential geometry. 1]] |last2=Nomizu |first2=Katsumi |author-link2=w:Katsumi Nomizu |date=1996 |publisher=[[w:Wiley (publisher)|Wiley]] |isbn=978-0-471-15733-5 |place=New York}}</ref> <ref name="Tu11">{{Cite book |last=Tu |first=Loring W. |author-link=w:Loring W. 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journal |last1=Christensen |first1=Daniel |last2=Wu |first2=Enxin |date=2016 |title=Tangent spaces and tangent bundles for diffeological spaces |url=https://cahierstgdc.com/wp-content/uploads/2017/11/ChristensenWu.pdf |journal=[[w:Cahiers de Topologie et Géométrie Différentielle Catégoriques|Cahiers de Topologie et Géométrie Différentielle Catégoriques]] |volume=57 |issue=1 |pages=3–50 |arxiv=1411.5425}}</ref> <ref name="Bloh24">{{Citation |last=Blohmann |first=Christian |chapter=Elastic diffeological spaces |date=2024 |title=Recent advances in diffeologies and their applications |volume=794 |issue= |pages=49–86 |editor-last=Magnot |editor-first=Jean-Pierre |chapter-url=https://www.ams.org/books/conm/794/ |access-date=2025-03-17 |series=Contemporary Mathematics |place= |publisher=[[w:American Mathematical Society|American Mathematical Society]] |arxiv=2301.02583 |doi=10.1090/conm/794 |isbn=978-1-4704-7254-2}}</ref> <ref name="Nest21">{{Cite journal |last=van der Schaaf |first=Nesta |date=2021 |title=Diffeological Morita Equivalence |url=https://cahierstgdc.com/wp-content/uploads/2021/04/Van-der-Schaaf-_LXII-2.pdf |journal=[[w:Cahiers de Topologie et Géométrie Différentielle Catégoriques|Cahiers de Topologie et Géométrie Différentielle Catégoriques]] |volume=LXII |issue=2 |pages=177–238 |arxiv=2007.09901}}</ref> <ref name="GurIgl19">{{Cite journal |last1=Gürer |first1=Serap |last2=Iglesias-Zemmour |first2=Patrick |date=2019 |title=Differential forms on manifolds with boundary and corners |journal=[[w:Indagationes Mathematicae|Indagationes Mathematicae]] |volume=30 |issue=5 |pages=920–929 |doi=10.1016/j.indag.2019.07.004 |doi-access=free}}</ref> <ref name="Hain79">{{Cite journal |last=Hain |first=Richard M. |date=1979 |title=A characterization of smooth functions defined on a Banach space |url=https://www.ams.org/proc/1979-077-01/S0002-9939-1979-0539632-8/ |journal=[[w:Proceedings of the American Mathematical Society|Proceedings of the American Mathematical Society]] |volume=77 |issue=1 |pages=63–67 |doi=10.1090/S0002-9939-1979-0539632-8 |issn=0002-9939 |doi-access=free|url-access=subscription }}</ref> <ref name="Los92">{{Cite journal |last=Losik |first=Mark |date=1992 |title=О многообразиях Фреше как диффеологических пространствах |trans-title=Fréchet manifolds as diffeological spaces |url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ivm&paperid=4812&option_lang=eng |journal=Izv. Vyssh. Uchebn. Zaved. Mat. |language=ru |volume=5 |pages=36–42 |via=[[All-Russian Mathematical Portal]]}}</ref> <ref name="Los94">{{Cite journal |last=Losik |first=Mark |date=1994 |title=Categorical differential geometry |url=http://www.numdam.org/item/CTGDC_1994__35_4_274_0/ |journal=[[w:Cahiers de Topologie et Géométrie Différentielle Catégoriques|Cahiers de Topologie et Géométrie Différentielle Catégoriques]] |volume=35 |issue=4 |pages=274–290}}</ref> <ref name="FrolKrieg88">{{Cite book |last1=Frölicher |first1=Alfred |author-link1=w:Alfred Frölicher |title=Linear spaces and differentiation theory |last2=Kriegl |first2=Andreas |date=1988 |publisher=[[w:Wiley (publisher)|J. Wiley & sons]] |isbn=978-0-471-91786-1 |series=Pure and applied mathematics |location=Chichester}}</ref> <ref name="Miy25">{{Citation |last=Miyamoto |first=David |title=Lie algebras of quotient groups |date=2025-02-14 |url=https://arxiv.org/abs/2502.10260 |access-date=2025-03-21 |publisher= |arxiv=2502.10260 |id=}}</ref> <ref name="IglKarZad10">{{Cite journal |last1=Iglesias-Zemmour |first1=Patrick |last2=Karshon |first2=Yael |last3=Zadka |first3=Moshe |date=2010 |title=Orbifolds as diffeologies |url=https://www.ams.org/journals/tran/2010-362-06/S0002-9947-10-05006-3/S0002-9947-10-05006-3.pdf |journal=[[w:Transactions of the American Mathematical Society|Transactions of the American Mathematical Society]] |volume=362 |issue=6 |pages=2811–2831 |doi=10.1090/S0002-9947-10-05006-3 |jstor=25677806 |s2cid=15210173}}</ref> <ref name="Miy24">{{Citation |last=Miyamoto |first=David |title=Lie groupoids determined by their orbit spaces |date=2024-03-22 |url=https://arxiv.org/abs/2310.11968 |access-date=2025-03-21 |publisher= |arxiv=2310.11968 |id=}}</ref> <ref name="DonIgl85">{{Cite journal |last1=Donato |first1=Paul |last2=Iglesias |first2=Patrick |date=1985 |title=Exemples de groupes difféologiques: flots irrationnels sur le tore |trans-title=Examples of diffeological groups: irrational flows on the torus |journal=[[w:Comptes rendus de l'Académie des Sciences|C. R. Acad. Sci. Paris Sér. I]] |language=fr |volume=301 |issue=4 |pages=127–130 |mr=799609}}</ref> <ref name="KarMiyWat24">{{Cite journal |last1=Karshon |first1=Yael |last2=Miyamoto |first2=David |last3=Watts |first3=Jordan |date=2024-10-01 |title=Diffeological submanifolds and their friends |url=https://linkinghub.elsevier.com/retrieve/pii/S0926224524000639 |journal=Differential Geometry and Its Applications |volume=96 |article-number=102170 |arxiv=2204.10381 |doi=10.1016/j.difgeo.2024.102170 |issn=0926-2245}}</ref> <ref name="Jor82">{{Cite journal |last=Joris |first=Henri |date=1982-09-01 |title=Une C∞-application non-immersive qui possède la propriété universelle des immersions |trans-title=A non-immersive C∞-map which possesses the universal property of immersions |url=https://doi.org/10.1007/BF01899535 |journal=[[w:Archiv der Mathematik|Archiv der Mathematik]] |language=fr |volume=39 |issue=3 |pages=269–277 |doi=10.1007/BF01899535 |issn=1420-8938|url-access=subscription }}</ref> <ref name="Ham82">{{cite journal |last1=Hamilton |first1=Richard S. |author-link=w:Richard S. Hamilton |title=The inverse function theorem of Nash and Moser |url = https://www.ams.org/journals/bull/1982-07-01/S0273-0979-1982-15004-2/S0273-0979-1982-15004-2.pdf |journal=[[w:Bulletin of the American Mathematical Society|Bulletin of the American Mathematical Society]] |date=1982 |volume=7 |issue=1 |pages=65–222 |doi=10.1090/S0273-0979-1982-15004-2}}</ref> <ref name="Wal12">{{cite journal |last1=Waldorf |first1=Konrad |title=Transgression to loop spaces and its inverse, I: Diffeological bundles and fusion maps |journal=[[w:Cahiers de Topologie et Géométrie Différentielle Catégoriques|Cahiers de Topologie et Géométrie Différentielle Catégoriques]] |date=2012 |volume=53 |issue=3 |pages=162–210 |issn=2681-2363}}</ref> <ref name="Wat17">{{cite journal|last=Watts|first=Jordan|date=2017-02-01|title=The differential structure of an orbifold|url=https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-47/issue-1/The-differential-structure-of-an-orbifold/10.1216/RMJ-2017-47-1-289.full|journal=[[w:Rocky Mountain Journal of Mathematics|Rocky Mountain Journal of Mathematics]]|volume=47|issue=1|doi=10.1216/RMJ-2017-47-1-289|issn=0035-7596}}</ref> <ref name="Tah26">{{cite journal|last=Taho|first=Masaki|date=2026-03-15|title=Tangent spaces of diffeological spaces and their variants|url=https://www.sciencedirect.com/science/article/pii/S0166864126000313|journal=[[w:Topology and its Applications|Topology and its Applications]]|volume=381|pages=109741|doi=10.1016/j.topol.2026.109741|issn=0166-8641}}</ref> }} getz3oucv1yupx9wvyxzzeuhtvygb6w 3 327012 2844126 2821276 2026-11-03T07:30:01Z MoochieMoocho 3113741 /* */ 2844126 wikitext text/x-wiki ==Yummy, pie...== :p ==To do== * [[Meme Theory and Semiotics]] * Fix pages with errors from [[:Category:Hatnote templates with errors]] * Fix CS1 errors at [[:Category:CS1 errors]] == Biography == @[[User:PieWriter|PieWriter]] I got confused when I saw the comments at Rfd, since the originator was not Wikiversity. Just to explain, I discussed this with Administrators as of how to incorporate this and how to use it. Since visual content appeals more to students than dull text, it looks like an idea to add questions like "Who invented what and what are the results" (just simply formulated). The biography should be expanded to meet the requirements. Feel free to contribute if you wish. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:48, 11 February 2026 (UTC) :@[[User:Harold Foppele|Harold Foppele]] Can you show me who you discussed it with diffs? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:00, 11 February 2026 (UTC) ::Since it is an email discussion, that would be inappropriate. But feel free to share your thoughts. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:47, 11 February 2026 (UTC) == Pppery == Are you and Pppery the same user ? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:15, 11 February 2026 (UTC) :@[[User:Pppery|Pppery]] Why don’t u answer that? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:44, 11 February 2026 (UTC) :: We are not. [[User:Pppery|Pppery]] ([[User talk:Pppery|discuss]] • [[Special:Contributions/Pppery|contribs]]) 23:44, 11 February 2026 (UTC) == Wikiversity scope, etc == It is very clear that Wikiversity is not Wikipedia. You have probably worked out that Wikiversity is not a place where I am normally active. I am an active [[w:en:AFC]] reviewer, and follow drafts to Commons where I patrol for files which are not licenced correctly to load to Commons. That hobby work has led me here. I appear unable to have an effect on the Wikiversity contributor who is treating this place as enWiki, and whose understanding of copyright law seems impossible to educate. I am grateful for your assistance in this endeavour. I am not sure of the processes here. They appear to be more relaxed than enWiki, and are most assuredly less relaxed than Commons. The areas where I feel able to judge, professor (etc) profiles and copyright, I feel those here who administer the system, albeit with subtly different titles, might jump in with firm guidance. The other content, the educational content, I am wholly unable to judge. I'm not sure what I am asking you to do, but I hoe that someone such as you, who has the administrative toolkit, might offer that firm education and guidance which seems to be required by our enthusiastic contributor. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 07:47, 17 February 2026 (UTC) :@[[User:Timtrent|Timtrent]] Thank you for your message and for taking the time to look into this. :Just to clarify, I’m not a curator/custodian on Wikiversity, so I don’t have access to any special tools beyond those of a regular contributor. That said, I’m totally agree with what you raised :You’re right that Wikiversity operates somewhat differently from Wikipedia and Wikimedia Commons, however copyright policies are quite similar ([[WV:Copyrights]]). :I have tried interacting with the user, but he just brushes me off, claiming that I am not a curator and thus implying my actions have no value. :One suggestion is that we could file a report at [[WV:Request custodian action]] about a possible warning/block of the user, so the user understands the seriousness of their action. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 07:54, 17 February 2026 (UTC) ::For some weird reason I thought you had the admin tools. ::The user is extremely pleasant, but just does not hear what is said. What I think is needed is for someone to brandish the mop and bucket and to inform them firmly where their path is mistaken. I see that as a precise, assertive, and friendly interaction prior to action. I can see a list of those here who have those rights, but I have no concept of whom to choose to ask (I don't quite feel as if formal action via a drama board is needed yet). ::I have double checked my file copyright thinking with [[w:en:User talk: Diannaa#Copyright advice at Wikiversity, please|an enWiki copyright expert]] who has confirmed all I have said regarding copyright. ::Would you mind choosing a suitable curator/custodian, please, and asking them for friendly and educational intervention? They will also be able to advise on scope, though Wikiversity is very clear on what it is not. If blocks have to happen I see that as a later phase. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 08:42, 17 February 2026 (UTC) :::Pinging two reliable ones, @[[User:Atcovi|Atcovi]] and @[[User:MathXplore|MathXplore]]. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:59, 17 February 2026 (UTC) ::::With good fortune and not a little diplomacy I think it is possible to educate this user into being a good citizen here. I hope sanctions are not needed. I think they have an abundance of good faith, and are simply having trouble converting their approach and thinking from the world of academe to the world of WMF. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 10:28, 17 February 2026 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], PieWriter!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:33, 24 March 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} :<nowiki>:)</nowiki>[[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:35, 24 March 2026 (UTC) == [[Wikiversity:Candidates for Curatorship/PieWriter]] == I've closed [[Wikiversity:Candidates for Curatorship/PieWriter]] as successful, and you've been given the curator rights. Congratulations! Please don't hesitate to ask any questions if you have any. BTW, please make sure to add your name to [[Wikiversity:Support staff]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:07, 27 March 2026 (UTC) :Thanks and will do! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:18, 27 March 2026 (UTC) Congratulations. I've added you as custodian. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:57, 21 May 2026 (UTC) :Thank you! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:54, 21 May 2026 (UTC) : Congrats. Reminder to update [[Wikiversity:Support staff]]. Note: {{u|Atcovi}} to mentor. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:11, 22 May 2026 (UTC) ::Will do! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:12, 22 May 2026 (UTC) == Test pages == Note that [[Fairy Rings/Database]] was not a test page, rather project page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:10, 29 March 2026 (UTC) :@[[User:Juandev|Juandev]] Oh, I didn’t notice that. I thought it was also a test page. Is it possible to undelete the page? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:00, 30 March 2026 (UTC) ::Yes it is, try it @[[User:PieWriter|PieWriter]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:24, 30 March 2026 (UTC) :::@[[User:Juandev|Juandev]] I tried but only custodians can undelete [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 10:55, 30 March 2026 (UTC) ::::I see, I am sorry, I dont have all rights for all flags in my mind. But now I see, it was not created in English, so lets leave it like that. Thank you for your time and dedication. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:43, 1 April 2026 (UTC) :::::Thanks! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:25, 1 April 2026 (UTC) == Displaying diffs on talk page == At some point I recall seeing you use a neat way of showing visual diffs in a discussion, with old on left and new on right. Do you remember how? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:17, 1 June 2026 (UTC) :@[[User:Jtneill|Jtneill]] Yeah, I did it with the help of a user script, https://en.wikipedia.org/wiki/User:NguoiDungKhongDinhDanh/FormattedEditRequest [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:22, 1 June 2026 (UTC) == [[Motivation and emotion/Book/2026]] == Thanks for helping out. However, please be careful because I've undone/reverted some of your edits where you removed user names from topics. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:01, 10 August 2026 (UTC) :Sorry about that, thanks for your help! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:07, 10 August 2026 (UTC) hxr4ljisk55cmrjva4iebc4gtpkxuko 2844127 2844126 2026-11-03T07:31:05Z PieWriter 3039865 Reverted edits by MoochieMoocho ([[:m:User:PieWriter/massundo.js|massundo]]) 2844127 wikitext text/x-wiki ==To do== * [[Meme Theory and Semiotics]] * Fix pages with errors from [[:Category:Hatnote templates with errors]] * Fix CS1 errors at [[:Category:CS1 errors]] == Biography == @[[User:PieWriter|PieWriter]] I got confused when I saw the comments at Rfd, since the originator was not Wikiversity. Just to explain, I discussed this with Administrators as of how to incorporate this and how to use it. Since visual content appeals more to students than dull text, it looks like an idea to add questions like "Who invented what and what are the results" (just simply formulated). The biography should be expanded to meet the requirements. Feel free to contribute if you wish. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 11:48, 11 February 2026 (UTC) :@[[User:Harold Foppele|Harold Foppele]] Can you show me who you discussed it with diffs? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:00, 11 February 2026 (UTC) ::Since it is an email discussion, that would be inappropriate. But feel free to share your thoughts. [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 12:47, 11 February 2026 (UTC) == Pppery == Are you and Pppery the same user ? [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:15, 11 February 2026 (UTC) :@[[User:Pppery|Pppery]] Why don’t u answer that? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:44, 11 February 2026 (UTC) :: We are not. [[User:Pppery|Pppery]] ([[User talk:Pppery|discuss]] • [[Special:Contributions/Pppery|contribs]]) 23:44, 11 February 2026 (UTC) == Wikiversity scope, etc == It is very clear that Wikiversity is not Wikipedia. You have probably worked out that Wikiversity is not a place where I am normally active. I am an active [[w:en:AFC]] reviewer, and follow drafts to Commons where I patrol for files which are not licenced correctly to load to Commons. That hobby work has led me here. I appear unable to have an effect on the Wikiversity contributor who is treating this place as enWiki, and whose understanding of copyright law seems impossible to educate. I am grateful for your assistance in this endeavour. I am not sure of the processes here. They appear to be more relaxed than enWiki, and are most assuredly less relaxed than Commons. The areas where I feel able to judge, professor (etc) profiles and copyright, I feel those here who administer the system, albeit with subtly different titles, might jump in with firm guidance. The other content, the educational content, I am wholly unable to judge. I'm not sure what I am asking you to do, but I hoe that someone such as you, who has the administrative toolkit, might offer that firm education and guidance which seems to be required by our enthusiastic contributor. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 07:47, 17 February 2026 (UTC) :@[[User:Timtrent|Timtrent]] Thank you for your message and for taking the time to look into this. :Just to clarify, I’m not a curator/custodian on Wikiversity, so I don’t have access to any special tools beyond those of a regular contributor. That said, I’m totally agree with what you raised :You’re right that Wikiversity operates somewhat differently from Wikipedia and Wikimedia Commons, however copyright policies are quite similar ([[WV:Copyrights]]). :I have tried interacting with the user, but he just brushes me off, claiming that I am not a curator and thus implying my actions have no value. :One suggestion is that we could file a report at [[WV:Request custodian action]] about a possible warning/block of the user, so the user understands the seriousness of their action. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 07:54, 17 February 2026 (UTC) ::For some weird reason I thought you had the admin tools. ::The user is extremely pleasant, but just does not hear what is said. What I think is needed is for someone to brandish the mop and bucket and to inform them firmly where their path is mistaken. I see that as a precise, assertive, and friendly interaction prior to action. I can see a list of those here who have those rights, but I have no concept of whom to choose to ask (I don't quite feel as if formal action via a drama board is needed yet). ::I have double checked my file copyright thinking with [[w:en:User talk: Diannaa#Copyright advice at Wikiversity, please|an enWiki copyright expert]] who has confirmed all I have said regarding copyright. ::Would you mind choosing a suitable curator/custodian, please, and asking them for friendly and educational intervention? They will also be able to advise on scope, though Wikiversity is very clear on what it is not. If blocks have to happen I see that as a later phase. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 08:42, 17 February 2026 (UTC) :::Pinging two reliable ones, @[[User:Atcovi|Atcovi]] and @[[User:MathXplore|MathXplore]]. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:59, 17 February 2026 (UTC) ::::With good fortune and not a little diplomacy I think it is possible to educate this user into being a good citizen here. I hope sanctions are not needed. I think they have an abundance of good faith, and are simply having trouble converting their approach and thinking from the world of academe to the world of WMF. [[User:Timtrent|Timtrent]] ([[User talk:Timtrent|discuss]] • [[Special:Contributions/Timtrent|contribs]]) 10:28, 17 February 2026 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], PieWriter!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:33, 24 March 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} :<nowiki>:)</nowiki>[[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:35, 24 March 2026 (UTC) == [[Wikiversity:Candidates for Curatorship/PieWriter]] == I've closed [[Wikiversity:Candidates for Curatorship/PieWriter]] as successful, and you've been given the curator rights. Congratulations! Please don't hesitate to ask any questions if you have any. BTW, please make sure to add your name to [[Wikiversity:Support staff]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:07, 27 March 2026 (UTC) :Thanks and will do! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:18, 27 March 2026 (UTC) Congratulations. I've added you as custodian. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:57, 21 May 2026 (UTC) :Thank you! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:54, 21 May 2026 (UTC) : Congrats. Reminder to update [[Wikiversity:Support staff]]. Note: {{u|Atcovi}} to mentor. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:11, 22 May 2026 (UTC) ::Will do! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:12, 22 May 2026 (UTC) == Test pages == Note that [[Fairy Rings/Database]] was not a test page, rather project page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:10, 29 March 2026 (UTC) :@[[User:Juandev|Juandev]] Oh, I didn’t notice that. I thought it was also a test page. Is it possible to undelete the page? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:00, 30 March 2026 (UTC) ::Yes it is, try it @[[User:PieWriter|PieWriter]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:24, 30 March 2026 (UTC) :::@[[User:Juandev|Juandev]] I tried but only custodians can undelete [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 10:55, 30 March 2026 (UTC) ::::I see, I am sorry, I dont have all rights for all flags in my mind. But now I see, it was not created in English, so lets leave it like that. Thank you for your time and dedication. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:43, 1 April 2026 (UTC) :::::Thanks! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:25, 1 April 2026 (UTC) == Displaying diffs on talk page == At some point I recall seeing you use a neat way of showing visual diffs in a discussion, with old on left and new on right. Do you remember how? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:17, 1 June 2026 (UTC) :@[[User:Jtneill|Jtneill]] Yeah, I did it with the help of a user script, https://en.wikipedia.org/wiki/User:NguoiDungKhongDinhDanh/FormattedEditRequest [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:22, 1 June 2026 (UTC) == [[Motivation and emotion/Book/2026]] == Thanks for helping out. However, please be careful because I've undone/reverted some of your edits where you removed user names from topics. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:01, 10 August 2026 (UTC) :Sorry about that, thanks for your help! [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:07, 10 August 2026 (UTC) tg96hphhvf2z686hnjagtn0j5mm27tq 10 330232 2844020 2843973 2026-11-02T20:45:26Z Jtneill 10242 # Use an example that a wider audience can relate to; less than 5% of the world population are university students 2844020 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unkacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings ## Quizzes should not have headings; they should be embedded like figures and tables <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unkacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings ## Quizzes should not have headings; they should be embedded like figures and tables <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> isvfdlzhnfvzdedqzq434f980ry04ia 2844021 2844020 2026-11-02T20:53:07Z Jtneill 10242 2844021 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unkacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unkacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> iqdmkhoxowp3b7uxy5ykrpa1ymsjxpf 2844030 2844021 2026-11-02T21:44:57Z Jtneill 10242 2844030 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> rpybdubt1lfgupgxy2se9twgnq8t3t3 2844033 2844030 2026-11-02T21:58:21Z Jtneill 10242 2844033 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> 2zyytp6a7zjbntkl8p9hiqgi255hgti 2844046 2844033 2026-11-02T22:41:16Z Jtneill 10242 (also known as the Oxford comma) 2844046 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> 3t6h04t0krxxfapewogetsq3r5cap61 2844062 2844046 2026-11-02T23:14:12Z Jtneill 10242 2844062 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/ of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> dxgut9q0f6eq6bzwkqim9fk6se3zje7 2844063 2844062 2026-11-02T23:15:43Z Jtneill 10242 #* more substantial engagement in social contribution 2844063 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/ of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> b11xy4z2xoy4c1lckvcn4e29v9n5215 2844068 2844063 2026-11-02T23:59:42Z Jtneill 10242 2844068 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/ of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> hn4eogh3li5ywt91z98b4shr24fvcyl 2844069 2844068 2026-11-03T00:00:20Z Jtneill 10242 2844069 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/ of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> ib5mlpy4mc72ajoy421tyoe5f4coqon 2844070 2844069 2026-11-03T00:07:01Z Jtneill 10242 2844070 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> fkdd141r4qncaabn48dnxljh4f1dzwf 2844151 2844070 2026-11-03T09:18:30Z Jtneill 10242 remove " (watch out for [[w:AI slop|AI slop]])" 2844151 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these copy edits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes OK use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]]—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* This chapter appears to be human-written #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity #* In some places, there is overreliance on genAI #* There may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]] #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [ these cop yedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably good scenario or case study; could be more compelling # Basic scenario or case study # The written expression for the scenario needs improvement to be of a professional standard # Use a more practical, real-life scenario; move review of research into a subsequent section # Use an example that a wider audience can relate to; less than 5% of the world population are university students # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Figure 1 lacks relevance to the scenario # The Figure 1 caption should explain more clearly how it is related to the scenario # Instead of a diagram, use <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long or vague or overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved # Convert the explanation of the problem from bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions are very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Good—key theories are quite explained and applied; minor to moderate areas for improvements # Reasonably good—relevant theories are selected, described, and applied, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Insufficient integration between the most relevant theory(ies) and the best research # Basic, formulaic # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # Take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is reasonably good, but appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]] ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Style – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see {{vague}}, {{explain}}, {{rewrite}} etc. tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections ## Integrate research rather than having it as a stand-alone section(s) ## Integrate (embed) learning features such as quizzes, without their own headings, rather than having them as stand-alone sections ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Style – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Style – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Style – Tables --> ## Table(s) ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Style – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent ### More copy editing is needed to fix formatting inconsistencies <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Internal link style --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> b8llo8o0zes165knn2hk50k4gsl61gq 0 330317 2844017 2834956 2026-11-02T20:39:03Z James500 297601 /* */ Add 2844017 wikitext text/x-wiki {{Bibliography}} This part of the [[Universal Bibliography]] is a bibliography of languages. World *Keith Brown and Sarah Ogilvie. Concise Encyclopedia of Languages of the World. Elsevier. 2009. [https://books.google.co.uk/books?id=F2SRqDzB50wC&pg=PP1#v=onepage&q&f=false] *Anatole V Lyovin, Brett Kessler and William R Leben. An Introduction to the Languages of the World. 2nd Ed: 2017: [https://books.google.co.uk/books?id=RQGTDQAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Asya Pereltsvaig. Languages of the World: An Introduction. 2012. [https://books.google.co.uk/books?id=8q06xer0vHkC&pg=PP1#v=onepage&q&f=false] *Merritt Ruhlen. A Guide to the World's Languages. Vol 1 (Classification). Stanford University Press. 1987. [https://books.google.co.uk/books?id=WAMbAAAAIAAJ] *Bernard Comrie. The World's Major Languages. 2nd Ed: 2009: [https://books.google.co.uk/books?id=9S0rDwAAQBAJ&pg=PP1#v=onepage&q&f=false]. *George L. Campbell and Gareth King. Compendium of the World's Languages. 3rd Ed: 2013: [https://books.google.co.uk/books?id=DWAqAAAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Martin D Joachim. Languages of the World: Cataloging Issues and Problems. 1993. [https://books.google.co.uk/books?id=6u18PtO0BoQC&pg=PP1#v=onepage&q&f=false] Origin *Roy Harris. Origin Of Language. 1996. [https://books.google.co.uk/books?id=386lU_0oUWoC&pg=PR3#v=onepage&q&f=false] *James R Hurford. Origins of Language: A Slim Guide. 2014. [https://books.google.co.uk/books?id=InTiAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Merritt Ruhlen. The Origin of Language: Tracing the Evolution of the Mother Tongue. 1994. [https://books.google.com/books?id=retrAAAAIAAJ] *Language Origin: A Multidisciplinary Approach. 1992. [https://books.google.co.uk/books?id=z_yPBAAAQBAJ&pg=PA1933#v=onepage&q&f=false] *Jürgen Trabant and Sean Ward (eds). New Essays on the Origin of Language. 2001. [https://books.google.co.uk/books?id=Pt501C6Zv94C&pg=PP1#v=onepage&q&f=false] *Claire Lefebvre, Bernard Comrie and Henri Cohen (eds). New Perspectives on the Origins of Language. Studies in Language Companion series, vol 144. [https://books.google.co.uk/books?id=S64bAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Denis Bouchard. The Nature and Origin of Language. 2013. [https://books.google.co.uk/books?id=4cRoAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Giorgio Fano. The Origins and Nature of Language. Indiana University Press. [https://books.google.com/books?id=fdlrAAAAIAAJ] *Jean Aitchison. The Seeds of Speech: Language Origin and Evolution. 1996. Canto Ed: 2000. [https://books.google.co.uk/books?id=68Y5gUavbzwC&pg=PP1#v=onepage&q&f=false] *Morris Swadesh. The Origin and Diversification of Language. 2006. 2017. [https://books.google.co.uk/books?id=klUPEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Prehistory *Rudolf Botha and Chris Knight (eds). The Prehistory of Language. 2009. [https://books.google.co.uk/books?id=36tLTfV_hLcC&pg=PP1#v=onepage&q&f=false] *G Révész. The Origins and Prehistory of Language. Longmans, Green and Co. [https://books.google.co.uk/books?id=GxRZAAAAMAAJ] History *Tore Janson. The History of Languages: An Introduction. 2012. [https://books.google.co.uk/books?id=pE2N7noPfEoC&pg=PP1#v=onepage&q&f=false] *Tore Janson. Speak: A Short History of Languages. 2002. [https://books.google.co.uk/books?id=mAgGOU2XmCAC&pg=PP1#v=onepage&q&f=false] *Nicholas Ostler. Empires of the Word: A Language History of the World. Preface dated 2004. [https://books.google.co.uk/books?id=Mz2kxr6v2X4C&pg=PP1#v=onepage&q&f=false] *Steven Roger Fischer. History of Language. 1999. [https://books.google.co.uk/books?id=5i1Ql7QQy0kC&pg=PP1#v=onepage&q&f=false] *A S Diamond. The History and Origin of Language. 1959: [https://books.google.co.uk/books?id=mjcGAQAAIAAJ]. Routledge Revivals. [https://books.google.co.uk/books?id=P5jiEAAAQBAJ&pg=PA1#v=onepage&q&f=false] *Henry Sweet. The History of Language. 1900. [https://books.google.co.uk/books?id=PC1GGpv7vlsC&pg=PR3#v=onepage&q&f=false] Social history *Peter Burke and Roy Porter (eds). The Social History of Language. 1987. [https://books.google.co.uk/books?id=oyRshxHVV5sC&pg=PP1#v=onepage&q&f=false] Story *Charles Barber. The Story of Language. Pan Books. 1964. [https://books.google.co.uk/books?id=gx0RAQAAIAAJ] *[[w:en:Mario Pei|Mario Pei]]. The Story of Language. 1949. Lippincott. Revised Ed: 1965. [https://books.google.co.uk/books?id=lqEviMzgv7wC]. Review: [https://books.google.co.uk/books?id=aaCvFv11ZJ4C 67] The Literary Guide 82 (May 1952) Classification *April McMahon and Robert McMahon. Language Classification by Numbers. 2005. [https://books.google.co.uk/books?id=CrEUDAAAQBAJ&pg=PP1#v=onepage&q&f=false] *CF and FM Voegelin. Classification and Index of the World's Languages. (Foundations of Linguistics series). Elsevier. New York. 1977. ISBN 0444001557. [https://books.google.co.uk/books?id=2LAuAAAAYAAJ] Extinct *Johannes Friedrich. Extinct Languages. 1957. [https://books.google.co.uk/books?id=SzcDAAAAMAAJ] *K David Harrison. When Languages Die: The Extinction of the World's Languages and the Erosion of Human Knowledge. 2007. [https://books.google.co.uk/books?id=GTfRCwAAQBAJ&pg=PP1#v=onepage&q&f=false] Dead *Coulter H George. How Dead Languages Work. 2020. [https://books.google.co.uk/books?id=xEfWDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Indo-European languages *Mate Kapović (ed). The Indo-European Languages. 2nd Ed: 2017: [https://books.google.co.uk/books?id=8i0lDwAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Anna Giacalone Ramat and Paolo Ramat (eds). The Indo-European Languages. 1998. [https://books.google.co.uk/books?id=vwUMNCYbLL0C&pg=PP1#v=onepage&q&f=false] **La Lingue Indoeuropee. 1993. *Philip Baldi. An Introduction to the Indo-European Languages. 1983. [https://books.google.co.uk/books?id=lq-mkL23oh8C&pg=PP1#v=onepage&q&f=false] *W B Lockwood. A Panorama of Indo-European Languages. 1972. [https://books.google.co.uk/books?id=QTLMEQAAQBAJ&pg=PA1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=xJ0cAQAAIAAJ] Anatolian languages *Donald C Swanson. A Select Bibliography of the Anatolian Languages. 1948. [https://books.google.co.uk/books?id=OR3KP8kCjzUC] Reprinted from Bulletin of the New York Public Library, [https://books.google.co.uk/books?id=ktkaAAAAMAAJ vol 52], nos 5 and 6, May and June 1948, pp 3 to 26. Hittite *Theo van den Hout. The Elements of Hittite. 2011. [https://books.google.co.uk/books?id=QDJNg5Nyef0C&pg=PR3#v=onepage&q&f=false] *Harry A Hoffner Jr and H Craig Melchert. A Grammar of the Hittite Language. [https://books.google.co.uk/books?id=Gq1QEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Edgar H Sturtevant and E Adelaide Hahn. A Comparative Grammar of the Hittite Language. 1951. [https://books.google.co.uk/books?id=5GRiAAAAMAAJ] *Jaan Puhvel. Hittite Etymological Dictionary. [https://books.google.co.uk/books?id=kghtOX_crPMC&pg=PP1#v=onepage&q&f=false] *Edgar H Sturtevant. A Hittite Glossary. 2nd Ed: 1936. Maltese *See [[w:mt:Bibljografija tal-lingwa Maltija]] Judaeo-Spanish (Ladino) *See [[w:lad:Vikipedya:Bibliografia del djudeo-espanyol]] Asian *Cliff Goddard. The Languages of East and Southeast Asia: An Introduction.2005. [https://books.google.co.uk/books?id=364UDAAAQBAJ&pg=PP1#v=onepage&q&f=false] South Asian *Kārumūri V Subbārāo. South Asian Languages: A Syntactic Typology. 2012. [https://books.google.co.uk/books?id=ZCfiGYvpLOQC&pg=PP1#v=onepage&q&f=false] *Veneeta Dayal and Anoop Mahajan. Clause Structure in South Asian Languages. 2004. [https://books.google.co.uk/books?id=puC-wWcl7tQC&pg=PP1#v=onepage&q&f=false] East Asian *Papers in East Asian Languages [https://books.google.co.uk/books?id=JIO5KcazJnYC] *Nam-kil Kim and Henry H Tiee. Studies in East Asian Linguistics. 1985. [https://books.google.co.uk/books?id=vxoaAQAAIAAJ] *Linguistic Interfaces in East-Asian Languages: A Festschrift in Honor of Yoshihisa Kitagawa. (Studies in East Asian Linguistics.) [https://books.google.co.uk/books?id=k8QYEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Benjamin A Elman (ed). Rethinking East Asian Languages, Vernaculars, and Literacies, 1000–1919. [https://books.google.co.uk/books?id=1Q6JBAAAQBAJ&pg=PR1#v=onepage&q&f=false] Chinese, Japanese and Korean *Reading in Asian Languages: Making Sense of Written Texts in Chinese, Japanese, and Korean. 2012. [https://books.google.co.uk/books?id=HZmpAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Insup Taylor and Martin M Taylor. Writing and Literacy in Chinese, Korean and Japanese. (Studies in Written Language and Literacy 3). 1995. [https://books.google.co.uk/books?id=WDw4gBaPjZgC&lpg=PP1&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=XW9IAAAAQBAJ&pg=PP1#v=onepage&q&f=false] Chinese and Japanese *Teruhiro Ishiguro and Kang Kwong Luke. Grammar in Cross-Linguistic Perspective: The Syntax, Semantics, and Pragmatics of Japanese and Chinese. (Linguistic insights, vol 57). Peter Lang. 2012. [https://books.google.co.uk/books?id=p1OqlAEACAAJ] Japan and Korea; Japanese and Korean *Nicolas Tranter (ed). The Languages of Japan and Korea. 2012. [https://books.google.co.uk/books?id=QB3DD8qSVnAC&pg=PP1#v=onepage&q&f=false] *Hajime Hoji (ed). Japanese/Korean Linguistics. 1990. vol 1. [https://books.google.co.uk/books?id=Suwm4WrQB6IC&pg=PP7#v=onepage&q&f=false] *Jieun Kiaer and Ben Cagan. Pragmatics in Korean and Japanese Translation. 2023. [https://books.google.co.uk/books?id=vnJ_EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *J Marshall Unger. The Role of Contact in the Origins of the Japanese and Korean Languages. University of Hawaii Press. 2009. [https://books.google.co.uk/books?id=sYULAQAAMAAJ] *Hisako Ueno and Jin Yu Young. [https://www.nytimes.com/2026/11/01/world/asia/japan-slang-korean-language.html New Slang in Japan Has an Unlikely Source: Korea]. The New York Times. 1 November 2026. Japonic *Michinori Shimoji. An Introduction to the Japonic Languages: Grammatical Sketches of Japanese Dialects and Ryukyuan Languages. Brill. 2022. [https://books.google.co.uk/books?id=TO77EAAAQBAJ&pg=PR1#v=onepage&q&f=false] *Yosuke Igarashi, Kenan Celik, Tatsuya Hirako and Hayato Aoi. Word-Prosodic Systems of Japonic Languages. Brill. 2026. [https://books.google.co.uk/books?id=B_3CEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Japanese and Ryukyuan *Moriyo Shimabukuro. The Accentual History of the Japanese and Ryukyuan Languages: A Reconstruction. 2007. [https://books.google.co.uk/books?id=n_V5DwAAQBAJ&pg=PR3#v=onepage&q&f=false] Japan *Masayoshi Shibatani. The Languages of Japan. CUP. 1990. [https://books.google.co.uk/books?id=sD-MFTUiPYgC&pg=PP1#v=onepage&q&f=false] *Handbook of Historical Japanese Linguistics [https://books.google.co.uk/books?id=xjz3EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Christopher Seeley. A History of Writing in Japan. 1991. [https://books.google.co.uk/books?id=KCZ2ya6cg88C&pg=PP1#v=onepage&q&f=false] Series *Handbooks of Japanese Language and Linguistics Ryukyuan *Handbook of the Ryukyuan Languages: History, Structure, and Use [https://books.google.co.uk/books?id=g_FeCAAAQBAJ&pg=PR3#v=onepage&q&f=false] Ainu *Handbook of the Ainu Language [https://books.google.co.uk/books?id=FAmKEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Katsunobu Izutsu. The Ainu Language: A Linguistic Introduction. Hokkaido University of Education. 2004. [https://books.google.co.uk/books?id=ty5kAAAAMAAJ] *Kirsten Refsing. The Ainu Language: The Morphology and Syntax of the Shizunai Dialect. 1986. [https://books.google.co.uk/books?id=LDJkAAAAMAAJ] *Batchelor. An Ainu-English-Japanese Dictionary. 1889: [https://books.google.co.uk/books?id=3gzhqi__TbEC&pg=PP7#v=onepage&q&f=false]. 2nd Ed: 1905: [https://archive.org/details/ainuenglishjapan00batcuoft/page/n4/mode/1up]. *Batchelor. A Grammar of the Ainu Language. 1903. [https://books.google.co.uk/books?id=G_xK9M0bOb8C] Japanese English *James Stanlaw. Japanese English: Language and Culture Contact. 2004.[https://books.google.co.uk/books?id=cdrHAQAAQBAJ&pg=PP1#v=onepage&q&f=false] ==Japanese== See [[Universal Bibliography/Languages/Japanese|Japanese]] ==Chinese== Chinese to English *英語大補帖~裝扮篇. Travelcom. (Taiwan). [https://books.google.co.uk/books?id=g7cWyH9k26EC&pg=PP1#v=onepage&q&f=false vol 9] [[Category:Languages]] b0wtt8i8jtla3quiloy5jxo9cn1vusp 0 330426 2844152 2830452 2026-11-03T09:21:46Z James500 297601 /* */ Add 2844152 wikitext text/x-wiki {{Bibliography}} This page is part of [[Universal Bibliography/Languages|bibliography of languages]]. This part of the [[Universal Bibliography]] is a bibliography of Japanese. Bibliography *Harald Suppanschitsch and Jürgen Stalph. Japanische Sprache und Schrift: eine Bibliographie des in deutscher Sprache veröffentlichten Schrifttums. 2001. [https://books.google.co.uk/books?id=7tBw_wLMOagC&pg=PR1#v=onepage&q&f=false] *Oskar Nachod. "Linguistics". Bibliography of the Japanese Empire 1906-1926. 1928. vol 2. Chapter XII. pp [https://archive.org/details/bibliographyofja0002oska/page/613/mode/1up 613] to 628, 753 and 754. *Wenckstern. "Philology: The Japanese Language". A Bibliography of the Japanese Empire. Chapter VI. vol 1, pp [https://books.google.co.uk/books?id=dcVAAAAAYAAJ&pg=PA74#v=onepage&q&f=false 74] to 88. vol 2, pp [https://archive.org/details/bibliographyofja0002frvo/page/74/mode/1up 74] to 89. General *Haruhiko Kindaichi. The Japanese Language. Tuttle. 1978. [https://books.google.co.uk/books?id=s_UZAQAAIAAJ] 1989. [https://books.google.co.uk/books?id=PdzkyasVMMoC] 2010. [https://books.google.co.uk/books?id=dAbRAgAAQBAJ&pg=PA1#v=onepage&q&f=false] *Osamu Mizutani. Japanese: The Spoken Language in Japanese Life. Japan Times. 1981. [https://books.google.co.uk/books?id=jZsPAAAAYAAJ] *Charles Berlitz. Passport to Japanese. 1985. [https://books.google.co.uk/books?id=MSQ04TeVfWYC] Periodicals *Japanese Language and Literature. (Journal of the Association of Teachers of Japanese.) [https://books.google.co.uk/books?&id=QpkmAQAAIAAJ] Periodical columns *[https://www.japantimes.co.jp/life/language/ Language]. [https://www.japantimes.co.jp/tag/bilingual/?type=column Bilingual]. [https://www.japantimes.co.jp/tag/vocabulary-boost/?type=column Vocabulary Boost]. The Japan Times. Kokugo *Paul H Clark. The Kokugo Revolution: Education, Identity, and Language Policy in Imperial Japan. (Japan Research Monograph 16). [https://books.google.co.uk/books?id=F6jSEQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Yeounsuk Lee. The Ideology of Kokugo: Nationalizing Language in Modern Japan. University of Hawaii Press. 2010. [https://books.google.co.uk/books?id=54wBEAAAQBAJ&pg=PP1#v=onepage&q&f=false] **Kokugo To Iu Shisō: Kindai Nihon No Gengo Ninshiki. (Japanese: 「国語」という思想: 近代日本の言語認識). Iwanami Shoten. Tokyo. 1996. Nihongo *Makoto Sugawara. Nihongo: A Japanese Approach to Japanese. East Publications. [https://books.google.co.uk/books?id=fKkPAAAAYAAJ] *Roy Andrew Miller. Nihongo: In Defence of Japanese. The Athlone Press. 1986. [https://books.google.co.uk/books?id=oRxkAAAAMAAJ] *Nihongo Notes. The Japan Times. [https://books.google.co.uk/books?id=LdkpAQAAIAAJ] *Yutaka Sato and Margaret Y. Yamashita. Nihongo: Introductory Japanese. 1994. vol 2. [https://books.google.co.uk/books?id=ptACuS6HnpUC&pg=PP1#v=onepage&q&f=false] *Minna No Nihongo I. 3A Corporation. (スリーエーネットワーク). 1998. [https://books.google.co.uk/books?id=G-bl2P5lRl4C&pg=PP1#v=onepage&q&f=false] **Minna No Nihongo II. [https://books.google.co.uk/books?id=4nHnMa4Zw-MC&pg=PP1#v=onepage&q&f=false] Introductions *A E Backhouse. The Japanese Language: An Introduction. Oxford University Press. 1993. [https://books.google.co.uk/books?id=vawPAAAAYAAJ] *Richard Bowring and Haruko Uryū Laurie. An Introduction to Modern Japanese. 1992. [https://books.google.co.uk/books?id=Gu3k3eiOXWAC&pg=PP1#v=onepage&q&f=false] Understanding *Yasuko Obana. Understanding Japanese: A Handbook for Learners and Teachers. 2000. [https://books.google.co.uk/books?id=I9IPAAAAYAAJ] Learn *Yuko Fukuroi. Learn Japanese. Institute of Asian Studies. 1997. [https://books.google.co.uk/books?id=0SJkAAAAMAAJ] *John Young and Kimiko Nakajima-Okano. Learn Japanese: New College Text: Volume IV. 1985. [https://books.google.co.uk/books?id=rxwxLVwW2t0C&pg=PP1#v=onepage&q&f=false] *John Young and Kimiko Nakajima-Okano. Learn Japanese: Pattern Approach. University of Maryland. 1963. [https://books.google.co.uk/books?id=pG1AsovGf3AC] *Nobuko Mizutani. Let's Learn Japanese. (Radio Japan). 1993. [https://books.google.co.uk/books?id=4urrPQAACAAJ] *Senko K Maynard. Learning Japanese for Real: A Guide to Grammar, Use, and Genres of the Nihongo World. University of Hawaii Press. 2011. [https://books.google.co.uk/books?id=QF4EEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Muneo Kimura. Learning Japanese: Techniques for Intermediate and Advanced Student. (Orientation Seminars on Japan, number 23). Office for the Japanese Studies Center, The Japan Foundation. 1986. [https://books.google.co.uk/books?id=ZyUHAQAAIAAJ] *Miwa Kai. Listen & Learn Japanese. 1959. Reprinted 1986. [https://books.google.co.uk/books?id=wBrYftZU6z4C&pg=PR1#v=onepage&q&f=false] Study *Jun Maeda. Let's Study Japanese. (Tuttle Language Library). 1st Ed: 1965. [https://books.google.co.uk/books?id=itdGCgAAQBAJ&pg=PP1#v=onepage&q&f=false] Courses *Fudeko Obazawa Reekie. A First Course in Japanese. 2007. [https://books.google.co.uk/books?id=VvmrFBsaXOkC&pg=PR3#v=onepage&q&f=false] *Intensive Course in Japanese. Language Services Co Ltd. [https://books.google.co.uk/books?id=SRhIAAAAMAAJ] [https://books.google.co.uk/books?id=0ytIAAAAMAAJ] *Akiyama. Nucleus Course in Japanese. Institute of Modern Languages. [https://books.google.co.uk/books?id=iGw-AAAAIAAJ] *Oreste Vaccari and Enko Elisa Vaccari. Complete Course of Japanese Conversation-Grammar. [https://books.google.co.uk/books?id=x9MTAQAAMAAJ] *Clay MacCauley. An Introductory Course in Japanese. 1897. [https://books.google.co.uk/books?id=Hmvl19e6ld4C&pg=PP5#v=onepage&q&f=false] Fundamentals *Toyoaki Uehara and Gisaburo N Kiyose. Fundamentals of Japanese. Indiana University Press. Bloomington and London. Tenri University Press. Tenri 1974. Reviews: [https://www.jstor.org/stable/598969] and Hiroshi Miyaji, "Book Reviews" (1976) [https://books.google.co.uk/books?id=EilkAAAAMAAJ 11] The Journal of the Association of Teachers of Japanese 106 (No 1: January 1976) Essential *Essential Japanese: Speak Japanese with Confidence. Tuttle. 2012. [https://books.google.co.uk/books?id=aJzTAgAAQBAJ&pg=PA1#v=onepage&q&f=false] *Lynne Strugnell. Essential Japanese. Berlitz. [https://books.google.co.uk/books?id=2vxBU3vjytQC] *Samuel E Martin. Essential Japanese: An Introduction to the Standard Colloquial Language. 1954. [https://books.google.co.uk/books?id=rx5kAAAAMAAJ] *Helmut Morsbach and Kazue Kurebayashi. Essential Japanese: A Guidebook to Language and Culture. Penguin Books.1990. ISBN 9780140101881. [https://books.google.co.uk/books?id=3rqgQ7zW3AsC] Ultimate *Ultimate Japanese **Suguru Akutsu. Ultimate Japanese: Advanced. 1998. [https://books.google.co.uk/books?id=7VV4RAAACAAJ]. Review: [https://books.google.co.uk/books?id=GnMqAQAAIAAJ 33] The Journal of the Association of Teachers of Japanese 111 (No 2: October 1999) Easy *Samuel E Martin. Easy Japanese: A Direct Learning Approach for Immediate Communication. 1st Ed: 1957. 2nd Ed: 1959. 3rd Ed: 1962. 4th Ed: 2006: [https://books.google.co.uk/books?id=CKHTAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Jack Seward. Easy Japanese. 1992. [https://books.google.co.uk/books?id=jQIraVXUxN0C] *Fumiko Koide. Easy Japanese. Nippon Kyooiku Kiki Fukyu Center Company. 1971. [https://books.google.co.uk/books?id=Q4JEAQAAMAAJ] *Emiko Konomi. Easy Japanese: Learn to Speak Japanese Quickly! [https://books.google.co.uk/books?id=mjtRDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Basic *Eriko Sato. Basic Japanese. [Practice Makes Perfect]. Premium 3rd Ed: 2023.[https://books.google.co.uk/books?id=JmeYEAAAQBAJ] *NTC's Basic Japanese. [https://books.google.co.uk/books?id=hLyZCKpa8jMC] *Samuel E. Martin and Eriko Sato. Basic Japanese: Learn to Speak Japanese in 10 Easy Lessons. Tuttle. [https://books.google.co.uk/books?id=F1RSDwAAQBAJ&pg=PA1#v=onepage&q&f=false] *Shoko Hamano and Takae Tsujioka. Basic Japanese: A Grammar and Workbook. 2011. [https://books.google.co.uk/books?id=l0fJAwAAQBAJ&pg=PP1#v=onepage&q&f=false] Demystified, Dummies *Eriko Sato. Japanese Demystified. 2008. [https://books.google.co.uk/books?id=Ak7AlXKi3pYC&pg=PR3#v=onepage&q&f=false] *Eriko Sato. Japanese For Dummies. 2002. [https://books.google.co.uk/books?id=Oi6lpE_NC-wC] Hiroko Chiba and Erik Sato. 3rd Ed. [https://books.google.co.uk/books?id=Gql7DwAAQBAJ&pg=PP1#v=onepage&q&f=false] Intermediate *Michael L Kluemper and Lisa Berkson. Intermediate Japanese Textbook. 2022. [https://books.google.co.uk/books?id=7hl2EAAAQBAJ&pg=PP1#v=onepage&q&f=false] **Intermediate Japanese Workbook. [https://books.google.co.uk/books?id=4qB-EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hiyaku: An Intermediate Japanese Course. 2011. [https://books.google.co.uk/books?id=9ZDtCQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Haruko Laurie and Richard Bowring. Cambridge Intermediate Japanese. 2002. [https://books.google.co.uk/books?id=E1wLAQAAMAAJ] *Yasuko Ito Watt and Richard Rubinger. Readers Guide to Intermediate Japanese: A Quick Reference to Written Expressions. 1998. [https://books.google.co.uk/books?id=S8ACEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Intermediate to advanced *The Routledge Intermediate to Advanced Japanese Reader. [https://books.google.co.uk/books?id=ZcMfEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Advanced *Noriko Ishihara and Magara Maeda. Advanced Japanese: Communication in Context. 2010. [https://books.google.co.uk/books?id=gmBQDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *An Introduction to Advanced Spoken Japanese. Inter-university Center for Japanese Language Studies. Delmer M Brown. 1987. [https://books.google.com/books?id=Og96QDPsx18C] For high school students; High school programs *Esther M T Sato, Loren I Shishido and Masako Sakihara. Japanese Now. 1982. [https://books.google.co.uk/books?id=k17cHllNfTAC&pg=PP1#v=onepage&q&f=false vol 1] **Esther M T Sato and Masako Sakihara. 1990. [https://books.google.co.uk/books?id=vufHtRpVZt4C&pg=PP1#v=onepage&q&f=false vol 4]. For scientists and engineers *Edward E. Daub, R Byron Bird and Nobuo Inoue. Basic Technical Japanese. 科学技術日本語の基礎. University of Wisconsin Press. 1990. [https://books.google.co.uk/books?id=oN23JJhjFpwC&pg=PP1#v=onepage&q&f=false] Readings *Joseph K Yamagiwa (ed). Readings in Japanese Language and Linguistics. University of Michigan Press. [https://books.google.co.uk/books?id=76wPAAAAYAAJ] Dictionaries See also [[w:en:Japanese dictionary]] and [[w:en:Category:Japanese dictionaries]] Phrasebooks *Toshiya Enomoto. 旅の指さし会話帳 21 Japan [English Edition/英語版]: 日本語. [https://books.google.co.uk/books?id=XCbxxkBY-hkC&pg=PP1#v=onepage&q&f=false] [Point and speak]. Vocabulary *Akira Miura. Essential Japanese Vocabulary. Tuttle. [https://books.google.co.uk/books?id=ZZvTAgAAQBAJ&pg=PA1#v=onepage&q&f=false] *Schaum's Outline of Japanese Vocabulary [https://books.google.co.uk/books?id=fV0NAAAACAAJ] *Carol and Nobuo Akiyama. Japanese Vocabulary. Barron's. 1991. [https://books.google.co.uk/books?id=7Aa6PAAACAAJ] Words *Akira Miura. Japanese Words & Their Uses. Charles E Tuttle. 1983. [https://books.google.co.uk/books?id=MVVzBgAAQBAJ&pg=PP1#v=onepage&q&f=false] Verbs *Complete Japanese Verb Guide. Tuttle. 1989. [https://books.google.co.uk/books?id=I_EPCwAAQBAJ&pg=PP1#v=onepage&q&f=false] *P Suski. Japanese Verbs. (Super Review). Research & Education Association. 2002. [https://books.google.co.uk/books?id=9t6oHZh5gecC&pg=PP1#v=onepage&q&f=false] *Naoko Chino. Japanese Verbs at a Glance. Kodansha International. 1996. [https://books.google.co.uk/books?id=-8AjAQAAIAAJ] *600 Basic Japanese Verbs. Tuttle. [https://books.google.co.uk/books?id=wZgdBAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Roland A Lange. 501 Japanese Verbs. Barron's. 1988. [https://books.google.co.uk/books?id=ANQXAAAAIAAJ] **201 Japanese Verbs. 1971. [https://books.google.co.uk/books?id=Dve2QgAACAAJ] *Rita Lampkin. Japanese Verbs and Essentials of Grammar: A Practical Guide to the Mastery of Japanese. 1995. [https://books.google.co.uk/books?id=P_CyQgAACAAJ] *Suski. Conjugation of Japanese Verbs in the Modern Spoken Japanese. 1942. [https://books.google.co.uk/books?id=SZIPAAAAYAAJ] *G F Verbeck. A Synopsis of All the Conjugations of the Japanese Verbs. 1887. [https://books.google.co.uk/books?id=jEJlAAAAIAAJ&pg=PA1#v=onepage&q&f=false] *Ready Conjugator of Japanese Verbs and Adjectives [https://books.google.co.uk/books?id=jrNDAQAAIAAJ] *Tadao Miyamoto. The Light Verb Construction in Japanese: The Role of the Verbal Noun. 1999. [https://books.google.co.uk/books?id=pHKVTctA-WwC&pg=PP1#v=onepage&q&f=false] Adjectives *Ann Tarumoto. Complete Japanese Adjective Guide. Tuttle. 2001. [https://books.google.co.uk/books?id=SIC4CgAAQBAJ&pg=PP1#v=onepage&q&f=false] Copula *Tomiko Narahara. The Japanese Copula: Forms and Functions. 2002: [https://books.google.co.uk/books?id=UKOHDAAAQBAJ&pg=PP1#v=onepage&q&f=false]. Reviews: [https://www.jstor.org/stable/25064474] [https://www.jstor.org/stable/44486751] [https://www.jstor.org/stable/4176893] Idioms *Kodansha's Dictionary of Basic Japanese Idioms. 2002. [https://books.google.co.uk/books?id=mQ5gyagWePMC&pg=PP1#v=onepage&q&f=false] *Nobuo Akiyama and Carol Akiyama. Japanese Idioms. Barron's. 1996. [https://books.google.co.uk/books?id=V5YPAAAAYAAJ] *Michael L Maynard and Senko K Maynard. 101 Japanese Idioms: Understanding Japanese Language and Culture Through Popular Phrases. 1993. [https://books.google.co.uk/books?id=HXI-Xvv5dMYC] Grammar *Stefan Kaiser, Yasuko Ichikawa, Noriko Kobayashi and Hilofumi Yamamoto. Japanese: A Comprehensive Grammar. 2001. 2nd Ed: 2013: [https://books.google.co.uk/books?id=vJH3CumpiZEC&pg=PP1#v=onepage&q&f=false]. *Naomi H McGloin, Mutsuko Endo Hudson, Fumiko Nazikian and Tomomi Kakegawa. Modern Japanese Grammar: A Practical Guide. 2014. [https://books.google.co.uk/books?id=qcdBDgAAQBAJ&pg=PA11#v=onepage&q&f=false] *Yuki Johnson. Fundamentals of Japanese Grammar. [https://books.google.co.uk/books?id=keIZAQAAIAAJ] *Kazuhiro Teruya. A Systemic Functional Grammar of Japanese. 2007. [https://books.google.co.uk/books?id=SJcqAQAAIAAJ] *Kimihiko Nomura. Japanese Grammar: The Connecting Point. 2010. [https://books.google.co.uk/books?id=I913EQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Schaum's Outline of Japanese Grammar [https://books.google.com/books?id=oZMYsHvuhXIC] *Masahiro Tanimori and Eriko Sato. Essential Japanese Grammar. Tuttle. [https://books.google.co.uk/books?id=CUXRAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Zeljko Cipris and Shoko Hamano. Making Sense of Japanese Grammar: A Clear Guide through Common Problems. 2002. [https://books.google.co.uk/books?id=GZ0BEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Carol Akiyama and Nobuo Akiyama. Pocket Japanese Grammar. 4th Ed: 2020: [https://books.google.co.uk/books?id=aga9DwAAQBAJ&pg=PP1#v=onepage&q&f=false] **Japanese Grammar. 3rd Ed. [https://books.google.co.uk/books?id=cO5wDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Harold G Henderson. Handbook of Japanese Grammar. 1945. 2011. [https://books.google.co.uk/books?id=NYEBAwAAQBAJ&pg=PP1#v=onepage&q&f=false] *W P Lehmann and Lloyd Faust. A Grammar of Formal Written Japanese. (Harvard-Yenching Institute Studies, vol 5). 1951. [https://books.google.co.uk/books?id=50s0AAAAIAAJ] Read; Reading *[[w:en:Eleanor Jorden|Eleanor Harz Jorden]] and Hamako Ito Chaplin. Reading Japanese. Yale University Press. 1976. [https://books.google.co.uk/books?id=1MF6kCogEx0C&pg=PP1#v=onepage&q&f=false] *Jiří Jelínek and Patricia A Heron. Reading Japanese: A self-instructional manual for beginners, leading to independent translating ability. University of Sheffield, Centre of Japanese Studies. 1975. [https://books.google.co.uk/books?id=IKQPAAAAYAAJ] *Dale P Crowley, with the assistance of Yoshiyuki Kawata and Yoko Kawata. Manual for Reading Japanese. University Press of Hawaii. Honolulu. 1972. [https://books.google.co.uk/books?id=nK0PAAAAYAAJ] *John Braden. Read Practical Japanese. Kenkyusha. 1976. [https://books.google.co.uk/books?id=3MAPAAAAYAAJ] *Setsuko Aihara, with Graham Parkes. Strategies for Reading Japanese: A Rational Approach to the Japanese Sentence. Japan Publications Trading Company. Tokyo. 1992. [https://books.google.co.uk/books?id=tMs_AQAAIAAJ] *Len Walsh. Read Japanese Today: The Easy Way to Learn Hiragana, Katakana and Kanji. Tuttle. [https://books.google.co.uk/books?id=1hjBEQAAQBAJ&pg=PP1#v=onepage&q&f=false]. Read Japanese Today: The Easy Way to Learn 400 Practical Kanji. [https://books.google.co.uk/books?id=QcrXBQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Arthur Rose-Innes. Japanese Reading for Beginners. K Yoshikawa & Co. [https://books.google.co.uk/books?id=cP1z4IcbiO4C] Slang *[https://www.japantimes.co.jp/life/2026/07/27/language/genz-slang-mlk-monaki/ J-pop idols are behind much of Japan’s Gen Z slang]. The Japan Times. 27 July 2026. See also *[[w:en:Category:Japanese language learning resources]] ==Writing== Written; Writing *David Ashworth and Ikumi Hitosugi. Written Japanese: An Introduction. 1993. [https://books.google.co.uk/books?id=fLDhgDHj7_EC&pg=PP1#v=onepage&q&f=false] *Heath Rose. The Japanese Writing System: Challenges, Strategies and Self-regulation for Learning Kanji. 2017. [https://books.google.co.uk/books?id=ZDU8DwAAQBAJ&pg=PA1924#v=onepage&q&f=false] *Basil Hall Chamberlain. A Practical Introduction to the Study of Japanese Writing. 1899. [https://books.google.co.uk/books?id=-SWFGQkuJN8C&pg=PP5#v=onepage&q&f=false] Handwritten *P G O'Neill. A Reader of Handwritten Japanese. Kodansha International. 1984. [https://books.google.co.uk/books?id=r-MZAQAAIAAJ] ==Characters and syllabaries== *Andrew N Nelson. Modern Reader's Japanese-English Character Dictionary. 1962. 1st Revised Ed: 1966. 2nd Revised Ed: 1974. Classic Ed: 1995. [https://books.google.co.uk/books?id=fKuHCgAAQBAJ&pg=PP1#v=onepage&q&f=false]. Review: The Incorporated Linguist, [https://books.google.co.uk/books?id=E2MtAQAAMAAJ vol 5], no 1, p 24. **John H Haig. The Compact Nelson: Japanese - English Character Dictionary. Tuttle. 1999. [https://books.google.co.uk/books?id=3bzewAEACAAJ] *Mark Spahn and Wolfgang Hadamitzky. Japanese Character Dictionary: With Compound Lookup via Any Kanji. 漢英熟語リバ一ス字典. Nichigai Associates. 1989. [https://books.google.co.uk/books?id=RqEPAAAAYAAJ] *NTC's New Japanese-English Character Dictionary. 1993. ISBN 0844284343. [https://books.google.co.uk/books?id=rZoFwQEACAAJ] **New Japanese-English Character Dictionary. Kenyusha. Tokyo. 1990. ISBN 4767490405. Hiragana and katakana *Timothy G Stout. Japanese Hiragana & Katakana for Beginners. 2011. [https://books.google.co.uk/books?id=ZPs8BAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Kenneth G Henshall and Tetsuo Takagaki. Learning Japanese Hiragana and Katakana. Revised 2nd Ed. [https://books.google.co.uk/books?id=QyfRAgAAQBAJ&pg=PP1#v=onepage&q&f=false] **Guide to Learning Hiragana & Katakana. Tuttle. 1990. [https://books.google.com/books?id=18i1QgAACAAJ] *Glen McCabe. Japanese Hiragana and Katakana Flash Cards. 2012. [https://books.google.co.uk/books?id=aSFFBAAAQBAJ&pg=PA34#v=onepage&q&f=false] *Richard S Keirstead. Japanese Hiragana & Katakana: Language Practice Pad. 2016. [https://books.google.co.uk/books?id=yPxHDgAAQBAJ&pg=PP1#v=onepage&q&f=false] Hiragana *Fujihiko Kaneda. Easy Hiragana. Passport Books. 1989. [https://books.google.com/books?id=CvQZAQAAIAAJ] *James W Heisig. Remembering the Hiragana: a complete course on how to teach yourself the Japanese syllabary in 3 hours. Japan Publications Trading Co. 1987. [https://books.google.co.uk/books?id=VdEPAAAAYAAJ] *Timothy G Stout. Japanese Hiragana for Beginners. 2007. [https://books.google.co.uk/books?id=dR_RAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Jim Gleeson. Writing Japanese Hiragana: An Introductory Japanese Language Workbook. 1996. Revised Ed: 2015: [https://books.google.co.uk/books?id=YtZGCgAAQBAJ&pg=PP1#v=onepage&q&f=false]. *Yuko Green. My First Hiragana Activity Book. 2000. [https://books.google.co.uk/books?id=C-OKxX_cdpgC&pg=PP1#v=onepage&q&f=false] Katakana *Tina Wells. Easy Katakana: How to Read and Write English Words Used in Japanese. Passport Books. 1989. [https://books.google.co.uk/books?id=-ZSDP-9i9oUC] *Helmut Morsbach, Kazue Kurebayashi and James W. Heisig. Remembering the Katakana. Japan Publications Trading Co. 1990. [https://books.google.co.uk/books?id=HeAPAAAAYAAJ] *Jim Gleeson. Writing Japanese Katakana: An Introductory Japanese Language Workbook. 1996. Revised Ed: 2015: [https://books.google.co.uk/books?id=rNZGCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Timothy G Stout. Japanese Katakana for Beginners: First Steps to Mastering the Japanese Writing System. 2007. [https://books.google.co.uk/books?id=W5sdBAAAQBAJ&pg=PP1#v=onepage&q&f=false] Kanji and kana *Wolfgang Hadamitzky and Mark Spahn. Japanese Kanji and Kana: A Complete Guide to the Japanese Writing System. 1981. 2nd Ed: 1997. 3rd Ed: 2011. [https://books.google.co.uk/books?id=3w7QAgAAQBAJ&pg=PP1#v=onepage&q&f=false] Kanji *James W Heisig. Remembering the Kanji. 1977. 5th Ed: 2007. [https://books.google.co.uk/books?id=TtEaylKrGaMC&pg=PP1#v=onepage&q&f=false vol 1]. Remembering the Kanji 1. [https://books.google.co.uk/books?id=PYOUEAAAQBAJ&pg=PP1#v=onepage&q&f=false] **James W Heisig. Remembering the Kanji: A systematic guide to reading Japanese characters. 1987. [https://books.google.co.uk/books?id=IKQPAAAAYAAJ] *James W Heisig and Tanya Sienko. Remembering the Kanji 3. 1994. 2nd Ed: 2008. [https://books.google.co.uk/books?id=wTZ4x_BHe5EC&pg=PP1#v=onepage&q&f=false] *Editorial staff of The East magazine. Kanji Kanji. The East Publications Inc. Tokyo. 1972: [https://books.google.co.uk/books?id=HcYPAAAAYAAJ]. Revised Ed: 1983: [https://books.google.co.uk/books?id=T8YPAAAAYAAJ]. *Andrew Dykstra. Kanji 1-2-3. Kanji Press. [https://books.google.co.uk/books?id=SnBWAAAAYAAJ] *The Learner's Japanese Kanji Dictionary. Tuttle. *Naoomi Kuratani, Akemi Kobayashi and Shunsuke Okunishi (eds). A New Dictionary of Kanji Usage あたらしい漢字用法辞典. Gakken. 1982. [https://books.google.co.uk/books?id=5-C4AAAAIAAJ]. Review: "The Slimline Kanji Dictionaries" (1996) 9 International Journal of Lexicography 132 (No 2: June). Abstracts: [https://books.google.co.uk/books?id=2vYPAQAAMAAJ] [https://academic.oup.com/ijl/article-abstract/9/2/132/930154] *Jack Halpern. The Kodansha Kanji Usage Guide: An A to Z of Kun Homophones. [https://books.google.co.uk/books?id=qgWQEAAAQBAJ] *Laurence Matthews. Kanji Fast Finder 漢字早引き辞典. Tuttle. [https://books.google.co.uk/books?id=7SdpPwAACAAJ] *Glen Nolan Grant. Mastering Japanese Kanji. Tuttle. 2009. [https://books.google.co.uk/books?id=0L1GCgAAQBAJ&pg=PP1#v=onepage&q&f=false vol 1]. *John Millen. Kanji Power: A Workbook for Mastering Japanese Characters. Tuttle Publishing. 1993. [https://books.google.co.uk/books?id=uPu4AAAAIAAJ] *Oreste Vaccari. Standard Kanji. Revised Ed. 1949. [https://books.google.co.uk/books?id=b1Q98sCcgV0C] *P G O'Neill. Essential Kanji. Weatherhill. 1973 [https://books.google.co.uk/books?id=VYadVK-DqSYC]. Paperback Ed: 1987: [https://books.google.co.uk/books?id=dQ25AAAAIAAJ]. *Essential Japanese Kanji. Tuttle. 2015. [https://books.google.co.uk/books?id=Gr5GCgAAQBAJ&pg=PP1#v=onepage&q&f=false vol 1]. 2016. [https://books.google.co.uk/books?id=F8A0DwAAQBAJ&pg=PP1#v=onepage&q&f=false vol 2]. *Kodansha's Compact Kanji Guide: A new character dictionary for students and professionals. Kodansha International. 1991. Review: Gerald B Mathias and Timothy J Vance (1992) [https://books.google.co.uk/books?id=2loLAQAAMAAJ 26] The Journal of the Association of Teachers of Japanese 47 (No 1: April 1992) *Fujihiko Kaneda. Easy Kanji. Passport Books. 1996. [https://books.google.co.uk/books?id=R7ZM9Dao7NMC] *Erik Sato. Learning Japanese Kanji: Practice Book. Tuttle. [https://books.google.co.uk/books?id=IcA0DwAAQBAJ&pg=PP1#v=onepage&q&f=false vol 2]. *The Second 100 Japanese Kanji. Tuttle. [https://books.google.co.uk/books?id=gUjRAgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Yasuko Kosaka Mitamura and Joyce Yumi Mitamura. Let's Learn Kanji: An Introduction to Radicals, Components and 250 Very Basic Kanji. Kodansha International. 1997. [https://books.google.co.uk/books?id=khnrnBXLciIC&pg=PP1#v=onepage&q&f=false] **Richard Glenn Covington, Yasuko Kosaka Mitamura and Joyce Yumi Mitamura. Let's Learn More Kanji: Family Groups, Learning Strategies, and 300 Complex Kanji. [https://books.google.co.uk/books?id=HLEOAAAACAAJ] ==Linguistics== *Yoko Hasegawa (ed). The Cambridge Handbook of Japanese Linguistics. 2018. [https://books.google.co.uk/books?id=CC5RDwAAQBAJ&pg=PP1#v=onepage&q&f=false] Reviews: [https://www.jstor.org/stable/48828212] [https://online.ucpress.edu/jjs/article-abstract/46/2/536/210735/Review-The-Cambridge-Handbook-of-Japanese] *Shigeru Miyagawa and Mamoru Saito (eds). The Oxford Handbook of Japanese Linguistics. 2008. [https://books.google.co.uk/books?id=4CS07LRO8O8C&pg=PP1#v=onepage&q&f=false] *Natsuko Tsujimura. An Introduction to Japanese Linguistics. 1996. Reviews: [https://www.jstor.org/stable/489672] [https://www.jstor.org/stable/417343]. 3rd Ed. [https://books.google.co.uk/books?id=LdaYAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Yoko Hasegawa. Japanese: A Linguistic Introduction. 2015. [https://books.google.co.uk/books?id=gpeiBQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Toshiko Yamaguchi. Japanese Linguistics in Use: An Introduction for Language Learners. 2025. [https://books.google.co.uk/books?id=QP-YEQAAQBAJ&pg=PP1#v=onepage&q&f=false] **Japanese Linguistics: An Introduction. 2007. Review: [https://www.jstor.org/stable/25173131] **Japanese Language in Use: An Introduction. 2007. *Natsuko Tsujimura. Japanese Linguistics. 2005. [https://books.google.com/books?id=bgJ8PgAACAAJ] *Tetsuo Harada. Outlines of Modern Japanese Linguistics. Tateshina Print Company. 1966. [https://books.google.co.uk/books?id=jTcHAQAAIAAJ] Periodicals, Linguistics *Papers in Japanese Linguistics [https://books.google.co.uk/books?id=iZomAQAAIAAJ] *Journal of Japanese Linguistics [https://books.google.co.uk/books?id=458mAQAAIAAJ] Issues *Takashi Imai and Mamoru Saito (eds). Issues in Japanese Linguistics. (Studies in Generative Grammar 29). Foris Publications. Dordrecht, Holland. 1987. ISBN 90-6765-284-9. [https://books.google.co.uk/books?id=bYiFEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Yukio Otsu and Ann Farmer (eds). Theoretical Issues in Japanese Linguistics. (MIT Working Papers in Linguistics 2). 1980. [https://books.google.com/books?id=RDeBAAAAIAAJ] Kokugogaku and nihongogaku *Lidia Tanaka. "Japanese language studies: Kokugo as an ideology, nihongo as an autonomous and global scholarship?". Kaori Okano and Yoshio Sugimoto (eds). Rethinking Japanese Studies: Eurocentrism and the Asia-Pacific Region. 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[https://books.google.co.uk/books?id=_yBkAAAAMAAJ] Semantics and pragmatics *Wesley M Jacobsen and Yukinori Takubo (eds). Handbook of Japanese Semantics and Pragmatics. 2020. [https://books.google.co.uk/books?id=wUUCEAAAQBAJ&pg=PR3#v=onepage&q&f=false] *Elin McCready, Katsuhiko Yabushita and Kei Yoshimoto (eds). Formal Approaches to Semantics and Pragmatics: Japanese and Beyond. 2014. 2020. [https://books.google.co.uk/books?id=ZeBcBAAAQBAJ&pg=PR4#v=onepage&q&f=false] Morphology and phonology *Jeroen Maarten van de Weijer and Tetsuo Nishihara (eds). Issues in Japanese Phonology and Morphology. (Studies in Generative Grammar 51).  2001. [https://books.google.co.uk/books?id=G4p_t7jy28AC&pg=PP1#v=onepage&q&f=false] Phonetics and Phonology *Haruo Kubozono (ed). Handbook of Japanese Phonetics and Phonology. 2015. [https://books.google.co.uk/books?id=8vFeCAAAQBAJ&pg=PR3#v=onepage&q&f=false] *Hiromi Otaka. Phonetics and Phonology of Moras, Feet and Geminate Consonants in Japanese. 2009. [https://books.google.co.uk/books?id=39Q_AQAAIAAJ] *James D McCawley. The Phonological Component of a Grammar of Japanese. Mouton & Co NV. The Hague. 1968. [https://books.google.co.uk/books?id=3JoPAAAAYAAJ] Syntax *Masayoshi Shibatani, Shigeru Miyagawa and Hisashi Noda (eds). Handbook of Japanese Syntax. 2017. [https://books.google.co.uk/books?id=tk8_DwAAQBAJ&pg=PA1#v=onepage&q&f=false] *Nobuko Hasegawa. Japanese Syntax in Comparative Grammar. Kuroshio Publishers. Tokyo. 1993. [https://books.google.co.uk/books?id=ztApAQAAIAAJ] Phonetics *Daniel Lepetit and Reiko Makino. Japanese Phonetics: A Thematic Bibliography. Canadian Scholars. 1996. ISBN 1551300923. Catalogue: Canadian Books in Print: Author and Title Index 2005. [https://books.google.co.uk/books?id=q2WLJa9rY5MC&pg=PA1063#v=onepage&q&f=false p 1063]. *Society Newsletter. 1926 to 1996. [[w:ja:日本音声学会|The Phonetic Society of Japan]]. [https://www.psj.gr.jp/eng/publication/society-newsletter] **Journal of the Phonetic Society of Japan. 1997 onwards. [https://www.psj.gr.jp/eng/publication] *Tsutomu Akamatsu. Japanese Phonetics: Theory and Practice. Lincom Europa. 1997. [https://books.google.com/books?id=guUZAQAAIAAJ] *P M Suski. The Phonetics of Japanese Language: With Reference to Japanese Script. 1931: [https://books.google.co.uk/books?id=gpthAAAAMAAJ]. 2011: [https://books.google.co.uk/books?id=9DuiAwAAQBAJ&pg=PP1#v=onepage&q&f=false]. Phonology *Laurence Labrune. The Phonology of Japanese. 2012. [https://books.google.co.uk/books?id=ix9r6CbEl6IC&pg=PR3#v=onepage&q&f=false] *Tsutomu Akamatsu. Japanese Phonology: A Functional Approach. Lincom Europa. 2000. [https://books.google.com/books?id=R-QZAQAAIAAJ] *Mieko Shimizu Han. Japanese Phonology: An Analysis Based on Sound Spectrograms. Kenkyusha. 1962. [https://books.google.co.uk/books?id=T3Xl7SviXB4C] Pragmatics *Mutsuko Endo Hudson, Yoshiko Matsumoto, Junko Mori (eds). Pragmatics of Japanese: Perspectives on grammar, interaction and culture. 2018. [https://books.google.co.uk/books?id=2wZTDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Gabriele Kasper. Pragmatics of Japanese as Native and Target Language. Second Language Teaching & Curriculum Center, University of Hawaiʼi at Mānoa. 1992. [https://books.google.co.uk/books?id=2_8ifcjhpYQC] *Naoko Taguchi. Pragmatic Competence. (Mouton Series in Pragmatics). 2009. [https://books.google.co.uk/books?id=14dEa1EZm4oC&pg=PP1#v=onepage&q&f=false] Sociolinguistics *Roy Andrew Miller. The Japanese Language in Contemporary Japan: Some Sociolinguistic Observations. 1977. [https://books.google.co.uk/books?id=9RxkAAAAMAAJ] ==Translation== *Yoko Hasegawa. The Routledge Course in Japanese Translation. 2012. [https://books.google.co.uk/books?id=5kX1O4bCx_oC&pg=PP1#v=onepage&q&f=false]. Review: [https://www.jstor.org/stable/24394410] *Judy Wakabayashi. Japanese–English Translation: An Advanced Guide. 2021. [https://books.google.co.uk/books?id=Nqf7DwAAQBAJ&pg=PA1#v=onepage&q&f=false] ==Dialects and regions== Dialects *Nobuko Kibe, Tetsuo Nitta and Kan Sasaki (eds). Handbook of Japanese Dialects. 2025. [https://books.google.co.uk/books?id=8_Y9EQAAQBAJ&pg=PR1#v=onepage&q&f=false] Kansai *Peter Tse. Kansai Japanese: The Language of Osaka, Kyoto, and Western Japan. (Tuttle Language Library). 1993. Reprinted 2002. [https://books.google.co.uk/books?id=FvVkCwAAQBAJ&pg=PP1#v=onepage&q&f=false] *DC Palter and Kaoru Slotsve. Colloquial Kansai Japanese: The Dialects and Culture of the Kansai Region. 1995. [https://books.google.co.uk/books?id=rJEdBAAAQBAJ&pg=PP1#v=onepage&q&f=false] ==History== *Bjarke Frellesvig. A History of the Japanese Language. 2010. [https://books.google.co.uk/books?id=v1FcAgiAC9IC&pg=PP1#v=onepage&q&f=false] *Lone Takeuchi. The Structure and History of Japanese: From Yamatokotoba to Nihongo. 1999. [https://books.google.co.uk/books?id=sr8PAAAAYAAJ] *Ohno Susumu. The Origin of the Japanese Language. Kokusai Bunka Shinkokai. Tokyo. 1970. [https://books.google.co.uk/books?id=pqcPAAAAYAAJ] *N A Syromiatnikov. The Ancient Japanese Language. Nauka Publishing House. 1981. [https://books.google.co.uk/books?id=OB5kAAAAMAAJ] *Yaeko Sato Habein. The History of the Japanese Written Language. University of Tokyo Press. 1984. [https://books.google.co.uk/books?id=xh1kAAAAMAAJ] ==Old Japanese== *John R Bentley. A Descriptive Grammar of Early Old Japanese Prose. 2001. [https://books.google.co.uk/books?id=Eoqv_NcLJ4gC&pg=PP1#v=onepage&q&f=false] *Alexander Vovin. A Descriptive and Comparative Grammar of Western Old Japanese. 2005. [https://books.google.co.uk/books?id=Ba1xEQAAQBAJ&pg=PR1#v=onepage&q&f=false] 2nd Ed: 2020. [https://books.google.co.uk/books?id=xfP_DwAAQBAJ&pg=PR3#v=onepage&q&f=false vol 1]. ==Classical Japanese== Introduction *Akira Komai and Thomas H Rohlich. An Introduction to Classical Japanese. Bonjinsha. Tokyo. 1991. Review: "Textbook Review by Questionnaire" (1992) [https://books.google.co.uk/books?id=tClnAAAAMAAJ 26] The Journal of the Association of Teachers of Japanese 50 (No 1: April 1992) Grammar *Noriko Katsuki-Pestemer. A Grammar of Classical Japanese. Lincom Europa. 2009. [https://books.google.co.uk/books?id=PHoLAQAAMAAJ] *Haruo Shirane. Classical Japanese: A Grammar. 2005. [https://books.google.co.uk/books?id=M5-vVlcVEDkC&pg=PP1#v=onepage&q&f=false]. Review: [https://muse.jhu.edu/article/209495/summary] *Alexander Vovin. A Reference Grammar of Classical Japanese Prose. 2003. [https://books.google.co.uk/books?id=24GTVscUCX0C&pg=PP1#v=onepage&q&f=false] *Akira Komai. A Grammar of Classical Japanese. Culver Publishing. 1979. [https://books.google.co.uk/books?id=sL0PAAAAYAAJ] *Tadashi Ikeda. Classical Japanese Grammar Illustrated with Texts. The Toho Gakkai (The Institute of Eastern Culture). 1980. [https://books.google.co.uk/books?id=jQjUAAAAMAAJ] Dictionary *Ivan Morris. Dictionary of Selected Forms in Classical Japanese Literature. Columbia University Press. 1966. [https://books.google.co.uk/books?id=O70PAAAAYAAJ] *Jiří Jelínek. Classical Japanese-English Grammar Dictionary. University of Sheffield, Centre of Japanese Studies. 1976. [https://books.google.co.uk/books?id=qOEPAAAAYAAJ] Clauses *Stefan Kaiser. Circumnominal Relative Clauses in Classical Japanese: An Historical Study. Otto Harrassowitz. Wiesbaden. 1991. ISBN 344703212X. [https://books.google.co.uk/books?id=GHsPAAAAYAAJ] [[Category:Japanese]] sohvjcttd483b4yu7tzd3doi8a0eo3m 0 331035 2844071 2837316 2026-11-03T00:09:36Z Jtneill 10242 Copyediting 2844071 wikitext text/x-wiki {{title|Positive emotion dysregulation:<br>What is positive emotion dysregulation and how does it affect psychological functioning?}} __TOC__ ==Overview== {{RoundBoxTop|theme=8}} [[File:Woman in red sweater with hand in air.jpg|right|thumb|250px|'''Figure 1.''' Grace experiencing happiness and excitement following several positive life events.]] '''Scenario''' Grace is a university student who feels that she has just won the lottery of her life. First, she receives a PhD offer from her dream university. Shortly afterwards, she learns that she has been awarded a scholarship to support her studies. For the next several days, Grace feels unusually happy and energised (see Figure 1). She sleeps very little because she believes there is too much to achieve. To celebrate her success, she purchases an expensive laptop, makes an impulsive travel booking, and commits to several university projects despite having a full schedule. People close to Grace begin to wonder whether something is wrong. Her friends try to slow her down, but she ignores their concerns. From Grace's perspective, she is simply happy, motivated, and confident as different aspects of her life are finally coming together in a positive way. 💡 What is happening to Grace? Why have her positive emotions become so intense? {{RoundBoxBottom}} [[Motivation and emotion/Book/2020/Positive emotion|Positive emotions]] are a normal and important part of human life. Positive emotions encompass a variety of [[wikipedia:Feeling|feelings]] such as [[wikipedia:Joy|joy]], [[wikipedia:Interest_(emotion)|interest]], [[wikipedia:Love|love]], [[wikipedia:Contentment|contentment]], [[wikipedia:Gratitude|gratitude]], [[wikipedia:Awe|awe]], and [[wikipedia:Amusement|amusement]] (Campos et al., 2013). These emotions play an important role in expanding people's [[wikipedia:Thought|thoughts]] and possible [[wikipedia:Action_(philosophy)|actions]] to help them build lasting psychological and social resources (Fredrickson, 2001). However, positive emotions might not always be beneficial if they are inadequately regulated. Strong or prolonged positive emotion emotional states can sometimes interfere with adaptive functioning (Gruber et al., 2020). Positive emotion dysregulation helps to explain how positive emotional states can initially feel good but can sometimes contribute to harmful thoughts and behaviour. This chapter explores how positive emotions can shift from being adaptive to maladaptive, and how this can affect psychological functioning. {{RoundBoxTop|theme=2}} 🔎 '''Focus questions''' * What is positive emotion dysregulation, and how can it be distinguished from healthy positive emotions? * What factors contribute to difficulties in regulating positive emotions? * How does positive emotion dysregulation affect psychological functioning? {{RoundBoxBottom}} ==Positive emotion: From benefit to dysregulation== Understanding positive emotion dysregulation requires more than just identifying whether an emotional experience is pleasant or intense. [[Motivation and emotion/Book/2020/Positive emotion|Positive emotions]] such as joy, amusement, and interest are usually considered pleasant in nature (Fredrickson, 2001). These feelings shape the way people think, act, and connect with world around them. Positive emotion dysregulation refers to difficulties in responding to positive emotions (Mansueto et al., 2025). This involves problems such as accepting, controlling or managing actions while feeling happy or excited (Mansueto et al., 2025). Therefore, positive emotion dysregulation is not simply experiencing strong positive emotions. It is about struggling to regulate those feelings in a way that supports effective functioning. This section will consider why positive emotions are important, how they are normally regulated, and how difficulties in regulating them can lead to positive emotion dysregulation. === Positive emotions and its adaptive functions === [[Motivation and emotion/Book/2020/Positive emotion|Positive emotions]] play an important role in psychological functioning. They can influence cognition, behaviour, and social interactions (Campos et al., 2013). Instead of simply producing pleasant effects, positive emotions can encourage individuals to explore their environment, learn from experiences, engage with others and pursue meaningful goals. Positive emotions can broaden people's thoughts, cognitions, and behaviours (Fredrickson, 2001). [[Motivation and emotion/Book/2016/Broaden-and-build theory of positive emotions|Broaden-and-build theory]] by Fredrickson (2001) proposes that positive emotions broaden momentary thought-action repertoires that encourages exploration, flexibility, and engagement. For instance, excitement following major positive events may encourage students such as Grace to pursue new opportunities and get more involved in university. These positive emotions can help build psychological and social resources, including relationships and coping resources (Fredrickson, 2001). Nevertheless, a network analysis about broaden-and-build theory on two studies (''N'' = 312; ''N'' = 302) found support for relationships between positive emotions, resources and life outcomes (Roth et al., 2024). But the broadening component that broaden-and-build theory proposed consistently deviated from the theory’s predictions, questioning its validity (Roth et al., 2024). However, positive emotions are not always beneficial in every circumstance. Effects of positive emotions depend on intensity, timing, and context. This provides a foundation for understanding when positive emotion may become less adaptive (Gruber et al., 2011). This review challenges assumption of [[wikipedia:Positive_psychology|positive psychology]] that increasing positive emotion will improve well-being. It does not mean that strong positive emotions are always harmful. Gruber et al.'s (2020) review on positive emotion disturbance highlighted how positive emotion disturbance can interfere with adaptive functioning. Overall, these findings suggest that positive emotions can support adaptive functioning. But positive emotions are not automatic and are not the same for everyone. Kashdan et al. (2015) reviewed myths regarding positive emotion regulation. This article emphasised flexibility in how people regulate emotions. Emotional responses should be adjusted according to the situation and personal goals. Thus, healthy positive emotion is not defined by its intensity alone. It depends on how it can be regulated flexibly without interfering with functioning. === From emotion regulation to positive emotion dysregulation === [[File:Positive emotion dysregulation pathway.png|thumb|520x520px|'''Figure 2.''' Applying the emotion regulation process to a positive situation. ''Note''. Some parts of this image has been adapted from Gross (1998) process model of emotion regulation]] [[Emotional regulation]] refers to a person's ability to control their own emotional state (Gross, 2015). [[Motivation and emotion/Book/2026/Extended process model of emotion regulation|The process model of emotion]] [[Motivation and emotion/Book/2026/Extended process model of emotion regulation|regulation]] by Gross (1998) provides a framework for understanding how individual can modify emotional experiences (see Figure 2). This model proposes that regulation occurs at different stages, including situation selection, situation modification, attentional deployment, [[wikipedia:Cognitive_appraisal|cognitive appraisal,]] and response modulation. [[wikipedia:Emotional_self-regulation|Regulation]] can occur before an emotional response is fully developed or after an emotion has already been experienced. In Grace’s case, receiving the [[wikipedia:Doctor_of_Philosophy|PhD]] offer and scholarship shifted her attention towards positive achievement. She appraises the situation positively. According to Gross' (1998) process model, emotion regulation then influences how she responds to and manages these emotions (see Figure 2). It also shapes what she does next.   Successful regulation helps individuals modify their emotional experiences (Gross, 1998). In contrast, difficulties in managing emotional experiences can lead to [[Motivation and emotion/Book/2026/Emotion dysregulation|emotion dysregulation]]. Difficulties in regulating emotions can cause various interpersonal problems, [[wikipedia:Rumination_(psychology)|rumination]], and [[wikipedia:Mental_disorder|mental disorders]]. This highlights that it is important to understand how [[wikipedia:Emotion|emotions]] are experienced and how individuals respond to them. Importantly, positive emotions also require regulation. According to Carl et al. (2013), individuals may need to regulate their positive emotions based on their needs, goals, and circumstances. Research has heavily focused on negative emotions and regulation of positive emotion has received less attention. This provides a foundation to examine positive emotion dysregulation. === Recognising positive emotion dysregulation === Positive emotion dysregulation is defined as the difficulties in responding and managing positive emotional experiences (Weiss et al., 2015). It is a multifaced construct that involves difficulty in accepting positive emotions, maintaining goal-directed behaviour, and controlling behaviour when experiencing positive emotions (Weiss et al., 2015). These three elements provide a useful way to recognise difficulties in regulating positive emotions (see Table 1). An intense positive emotion does not necessarily mean dysregulated. Feelings such as happiness, excitement, or [[wikipedia:Enthusiasm|enthusiasm]] may be well-placed responses to certain life events. Context, intensity, persistence, and implications are crucial. Positive emotions can be considered [[wikipedia:Maladaptation|maladaptive]] if they are poorly fitted to the situation or if they start to interfere with functioning (Gruber et al., 2011). Positive emotion dysregulation occurs when these feelings are hard to manage and when they disturb behaviour, goals and daily life. In Grace’s scenario, she engages in impulsive behaviour and commits to projects that she will not be able to complete on time. This highlights dysregulation as it affects her daily functioning. A review by Vogel et al. (2023) suggests that there is a need to examine the challenges of regulating positive emotions rather than focusing exclusively on negative emotions. This review also highlights that dysregulation should be understood in relation to the feelings and the way a person behaves. Overall, recognising emotional dysregulation means looking past how strong the feeling is and focusing on how person can respond and manage that feeling. '''Table 1''' ''Key Elements For Recognising Positive Emotion Dysregulation'' ''Note. This table is based on the Difficulties in Emotion Regulation Scale-Positive (DERS-P) dimensions researched by Weiss et al. (2015).'' {| class="wikitable" style="margin: auto;" |- ! Dimension !Positive emotion regulation ! Positive emotion dysregulation |- | Accepting positive emotions |Accepts and acknowledges positive emotions without feeling that they are wrong or threatening | Feels uncomfortable or scared when experiencing positive emotions |- |Goal-directed behaviour |Can stay focused on goals while experiencing positive emotions |Difficulty in continuing goal-directed behaviour while experiencing positive emotions |- |Behavioural control |Can control behaviour and avoid acting impulsively when feeling happy or excited |Struggle in controlling behaviour or acts impulsively |} {{Robelbox|theme=5|title=Quiz}} [[Image: Hand Gesture - Holding a Magnifying Glass.gif|right|128px]] <quiz display=simple> Which statement best describes the difference between healthy positive emotions and positive emotion dysregulation? |type="()"} - Healthy positive emotions improves behaviour, whereas dysregulation occurs when positive behaviour becomes excessive - Healthy positive emotions means emotions are freely expressed, whereas dysregulation occurs when they are suppressed + Healthy positive emotions supports functioning and positive emotion dysregulation involves difficulty in managing positive emotions. - Healthy positive emotions are always beneficial and positive emotion dysregulation means positive emotions become negative. </quiz> {{Robelbox/close}} == Factors underlying positive emotion dysregulation == [[File:Positive emotion dysregulation.png|thumb|'''Figure 3.''' Factors that may contribute to positive emotion dysregulation ]] Positive emotion dysregulation is unlikely a result from a single factor. Instead, it is a result of biological, temperamental, coping, social and environment factors (see Figure 3). All these factors may interact to influence how strongly positive emotions are experienced and how they are regulated. This section examines biological and environmental influences, ways of coping and regulation of positive emotions, and environmental and social factors. === Biological and temperamental factors === One of the factors for positive emotion dysregulation can be an increased sensitivity to emotional events from a biological and temperamental perspective. Some people might be emotionally more sensitive or find it harder to control their emotional reactions. A {{ic|type of?}} review by Vogel et al. (2023) reported that [[wikipedia:Surgency|surgency]] {{explain}} negatively correlated with effortful control and self-regulation in childhood. This may suggest that high surgency can be associated with aggression and externalising behaviours. Nonetheless, surgency has positive implications too and its link with regulation changed at different developmental stages. Therefore, it is hard to make strong conclusion to explain dysregulation only through surgency. Difficulties may also arise if there is high emotional activation that exceeds individual’s cognitive capacity. Additionally, Vogel et al. (2023) review only highlighted that [[wikipedia:Temperament|temperamental]] traits represent a vulnerability but not the exact causes of positive emotion dysregulation. Linehan’s (1993) [[wikipedia:Biosocial_theory|biosocial model]] offers additional perspective where biological emotional sensitivity combined with environmental invalidation leads to emotional dysregulation. It results in difficulties in managing and responding to intense positive emotions. A limitation of this theory is that it is not designed for positive emotions specifically. But it can be used to predict that people with high biological emotional sensitivity will be affected by the way other people perceive their emotions. Taken together, the developmental findings by Vogel et al. (2023) combined with Linehan’s (1993) theory suggest that temperament may creates vulnerability. But the ability to regulate and control environmental experiences determine the impact of excessive positive emotion. === Coping and emotion-regulation strategies === A person's ability to cope with their emotions may determine whether their strong emotions remain adaptive or turn into something uncontrollable. The process model of emotion regulation by Gross (1998) provides a useful framework for understanding emotion regulation process. According to this model, regulation involves modifying emotional responses according to the situation and goals. Studies that look at emotion regulation support this idea. Livingstone and Srivastava (2012) found {{how}} three ways to increase the level of positive emotions which are engagement, betterment, and indulgence. This provides evidence that increasing emotions does not necessarily mean dysregulation. [[Motivation and emotion/Book/2020/Coping and emotion|Coping strategies]] may contribute to vulnerability to positive emotion dysregulation. To assess how regulation strategies work in real life conditions, Boemo et al. (2022) undertook a [[wikipedia:Systematic_review|systematic review]] and [[wikipedia:Meta-analysis|meta-analysis]] of experience sampling and daily diary designs. The findings showed that [[wikipedia:Appraisal_theory|reappraisal]] was associated with greater positive effect, while [[wikipedia:Rumination_(psychology)|rumination]], and suppression were associated with poorer affective outcomes. Reappraisal prospectively predicted positive affect (''r'' = .34), while rumination prospectively predicted negative affect (''r'' = .16). These results shows that how people manage emotions can influence emotional functioning. The review focused on emotion regulation, not on positive emotion dysregulation so it cannot prove that poor regulation leads to positive emotion dysregulation. Haag et al. (2024) conducted a systematic review of 11 studies involving 3,775 adolescents (''M'' = 13.4 years) to understand emotion regulation flexibility in adolescents. The results of this study found that greater emotion regulation flexibility is associated with more adaptive outcomes, including higher positive effect and self-worth. The study conceptualised emotion regulation flexibility in two ways: flexibility in regulating emotions according to the situation and the use of emotion regulation strategies. However, the evidence is limited as most studies were [[wikipedia:Cross-sectional_study|cross-sectional]] and community based. There was also heavy reliance on [[wikipedia:Self-report_study|self-report measures]]. Therfore, the findings suggests that poor regulation may increase vulnerability to emotional difficulties. But the study does not establish that low emotional regulation flexibility directly causes positive emotion dysregulation. Lastly, these studies combined provide a foundation to understand the relationship between the ability to cope with emotions and emotional dysregulation. But direct inferences cannot be made as research is broad and focused only on emotional dysregulation in general. Future research is needed to understand how to cope with intense positive emotions so that does not lead to positive emotion dysregulation. === Environmental and social factors === Social environmental factors may shape the way positive emotions are expressed and regulated. Goagoses et al. (2023) reviewed 27 studies on parenting and emotion dysregulation during childhood and adolescence. Results showed a small positive effect between negative parenting and emotion dysregulation (Δ = .20, ''p'' < .001). Additionally, there was a negative effect between positive parenting and emotion dysregulation (Δ = - .17, ''p'' < .001). But there was notable heterogeneity in both meta-analyses, meaning that results should be interpreted carefully. Furthermore, emotion dysregulation in this study was evaluated in general and not separated into positive and negative emotion dysregulation. Cultural norms might play a role in the expression of positive emotions. Huwaë and Schaafsma (2016) conducted research on positive emotion suppression in daily interactions. This study included 71 individuals of Chinese, Moluccas, and Dutch ancestry who resided in the Netherlands. Emotion suppression was greater in Chinese than in Dutch and Moluccas participants. These results suggest that culture and social relations can affect emotional expression. However, the small sample limits generalisability. Cultural expectations can discourage expressing positive emotions which may contribute to positive emotion dysregulation. Positive emotions can be influenced by peer interactions when regulating them. Miller-Slough & Dunsmore (2023) studied 30 adolescents aged 13-18 years. Results found that positive emotions were coached by parents rather than peers. Parental coaching was found to be linked to poor self-reported emotion regulation but good physiological regulation. These mixed findings suggest that social influences on positive emotion regulation are complex. Lastly, social factors affecting positive emotion dysregulation are multifaced. Similarly, Breaux et al. (2022) conducted a systematic review of parent emotion socialisation and positive emotions in child and adolescent samples. In seven included studies, the [[wikipedia:Correlation|correlational data]] suggested weak support for the association between emotion socialisation behaviour of parents and positive emotions. Intervention studies proved effective increasing both parenting behaviours and positive emotions in children. But these findings reflected emotion competence as a whole rather than the positive emotion dysregulation. Overall, social and environmental factors seem to be the contributing factor rather than direct cause of positive emotion dysregulation. Parenting could play a role in the development of emotion regulatory skills, but existing research cannot identify a clear association. {{Robelbox|theme=10|title=Quiz}}[[Image: Hand Gesture - Holding a Magnifying Glass.gif|right|128px]] <quiz display=simple> Why can high surgency not be considered a direct cause of positive emotion dysregulation? |type="()"} - Surgency only occurs during adulthood - Surgency has only negative effects - Surgency is not related to emotional functioning + The effects of surgency is adaptive and its association with regulation changes across development </quiz> {{Robelbox/close}} == Consequences of positive emotion dysregulation on psychological functioning == [[File:Consequences of positive emotion dysregulation.png|thumb|'''Figure 4.''' Positive emotion dysregulation and its potential consequences |400x400px]] The dysregulation of positive emotions may manifest itself through altered thinking processes, decision-making, and behaviours of an individual (see Figure 4). While positive emotions help expand attention and motivate individuals to explore more, strong and poorly regulated emotional activation may influence judgement. Gruber et al. (2020) proposed that positive emotions may become maladpative when they interfere with an individual's functioning. This section will discuss the way in which positive experiences may impact cognition, behaviour and decision-making processes. === Risk-taking and impulsivity === Positive emotion dysregulation may appear as reduced behavioural control specially when individuals experience intense positive emotions. Presence of positive emotional activation may lead to increased level of [[Motivation and emotion/Book/2017/Impulsivity|impulsivity]] and [[Motivation and emotion/Book/2010/Risk-taking motivation|risky behaviour]]. Positive urgency provides a precise reason for rash behaviour during high levels of positive emotions. Cyders and Smith (2008) define positive urgency as the tendency of behaving rashly during high levels of positive affect. This behavioural trend can be risky as well. However, positive urgency should not be equated with positive emotion dysregulation as it an impulsive trait itself and not an indicator of positive emotion dysregulation. A more direct study of positive emotion dysregulation and risk taking was conducted by Weiss et al. (2023). The research included 151 women from a community who suffered from intimate partner violence and were also consuming drugs and alcohol. Randomised participants underwent positive, negative, and neutral emotion inductions. Self-reported emotion dysregulation after positive emotion induction correlated with increased urges of substance use. Low levels of physiological regulation measured through high-frequency heart-rate variability were associated substance use and risk-taking. This provides evidence that difficulty in regulating positive emotions may contribute to reduced behavioural control. It also increases risk-taking behaviour during heightened positive emotional states. However, this specific clinical sample limits generalisability to the broader population. === Goal-directed functioning and cognitive control === Positive emotion dysregulation may affect goal-directed behaviour when intense positive emotions become difficult to manage. According to Weiss et al. (2015), impulse control and difficulties with goal-directed behaviour are the dimensions for positive emotion dysregulation. These difficulties arise when people experience positive emotions. This suggests that positive emotion dysregulation is not only characterised by the experience of strong positive emotions. It involves difficulty staying focused on goals and controlling behaviour. Nonetheless, the study was aimed at developing and validating the Difficulties in Emotion Regulation Scale-Positive (DERS-P). Hence, it cannot be concluded that positive emotion dysregulation causes difficulty with goal achievement. === Broader psychological functioning === The consequences of positive emotion dysregulation may extend beyond individual behaviours to broader mental health functioning. Weiss et al. (2020) compared the outcomes of 400 participants from the community who either experienced or had not experienced [[wikipedia:Psychological_trauma|trauma]] and [[Motivation and emotion/Book/2024/Post-traumatic stress disorder and emotion|post-traumatic stress disorder]] (PTSD). Positive emotion dysregulation was found to be higher in individuals who experienced trauma and had probable PTSD than those who experienced trauma but did not have probable PTSD. The effect size was moderate-to-large (''d'' = .66 to .73). This study showed importance of positive emotion regulation in functioning related to PTSD. However, the study used a cross-sectional design, thus could not prove the causality. Furthermore, Hechtman et al. (2013) also highlighted that further studies of positive emotion regulation in the context of [[wikipedia:Psychopathology|psychopathology]] is crucial. {{Robelbox|theme=4|title=Quiz}}[[Image: Hand Gesture - Holding a Magnifying Glass.gif|right|128px]] <quiz display=simple> Which of the following is correct regarding positive emotion dysregulation and psychological functioning? |type="()"} + Poorly regulated positive emotions can interfere with attention, wellbeing, relationships and everyday responsibilities - Positive emotions are always harmful - Positive emotions have no effect on behaviour and functioning - Intense positive emotions always indicate dysregulation </quiz> {{Robelbox/close}} ==Conclusion== Positive emotion dysregulation is not simply experiencing intense happiness, but difficulty in regulating positive emotions. Positive emotions support [[wikipedia:Motivation|motivation]], relationships, and overall wellbeing. In contrast, dysregulated positive emotions might be excessive, too persistent, and hard to control. The difficulties associated with dysregulation are a result of multiple factors. Some biological factors can be combined with strategies of emotion regulation and environmental factors to enhance vulnerability to positive emotion dysregulation. If positive emotions become hard to regulate, then it may affect everyday responsibilities and functioning. Positive emotion dysregulation can influence thoughts, [[wikipedia:Decision-making|decision-making]], goal-directed behaviour, and other mental health functioning. However, research on positive emotion dysregulation is still developing. The evidence is mostly correlational and population specific. Therefore, future longitudinal and experimental studies are needed to understand when positive emotions become dysregulated and how effective regulation can promote adaptive functioning. {{RoundBoxTop|theme=11}} [[File:Nuvola apps kuser.svg|Nuvola apps kuser|left|20px]] ''' Take-home messages:''' * Positive emotion dysregulation occurs when positive emotions are difficult to regulate in ways that are appropriate according to the situation. * When positive emotions are poorly regulated, they can interfere with people's wellbeing, attention, and everyday functioning. {{RoundBoxBottom}} ==See also == *[[Motivation and emotion/Book/2016/Broaden-and-build theory of positive emotions|Broaden-and-build theory of positive emotions]] (Book chapter, 2016) *[[Motivation and emotion/Book/2020/Coping and emotion|Coping and emotion]] (Book chapter, 2020) *[[Motivation and emotion/Book/2026/Emotion dysregulation|Emotion dysregulation]] (Book chapter, 2026) *[[wikipedia:Emotional_dysregulation|Emotional dysregulation]] (Wikipedia) *[[Motivation and emotion/Book/2020/Positive emotion|Positive emotion]] (Book chapter, 2020) ==References== {{Hanging indent|1= Boemo, T., Nieto, I., Vazquez, C., & Sanchez-Lopez, A. (2022). Relations between emotion regulation strategies and affect in daily life: A systematic review and meta-analysis of studies using ecological momentary assessments. ''Neuroscience & Biobehavioral Reviews'', ''139'', 104747. https://doi.org/10.1016/j.neubiorev.2022.104747 Breaux, R., Lewis, J., Cash, A. R., Shroff, D. M., Burkhouse, K. L., & Kujawa, A. (2022). Parent Emotion Socialization and Positive Emotions in Child and Adolescent Clinical Samples: A Systematic Review and Call to Action. ''Clinical Child and Family Psychology Review'', ''25''(1), 204–221. https://doi.org/10.1007/s10567-022-00388-2 Cyders, M. A., & Smith, G. T. (2008). Emotion-based dispositions to rash action: Positive and negative urgency. ''Psychological Bulletin'', ''134''(6), 807–828. https://doi.org/10.1037/a0013341 Campos, B., Shiota, M. N., Keltner, D., Gonzaga, G. C., & Goetz, J. L. (2013). What is shared, what is different? Core relational themes and expressive displays of eight positive emotions. ''Cognition and Emotion, 27''(1), 37–52. https://doi.org/10.1080/02699931.2012.683852 Carl, J. R., Soskin, D. P., Kerns, C., & Barlow, D. H. (2013). Positive emotion regulation in emotional disorders: A theoretical review. ''Clinical Psychology Review, 33''(3), 343–360. https://doi.org/10.1016/j.cpr.2013.01.003 Fredrickson B. L. (2001). The role of positive emotions in positive psychology. The broaden-and-build theory of positive emotions. ''The American psychologist, 56''(3), 218–226. https://doi.org/10.1037//0003-066x.56.3.218 Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology, 2''(3), 271–299. https://doi.org/https://doi.org/10.1037/1089-2680.2.3.271 Gruber, J., Mauss, I. B., & Tamir, M. (2011). A Dark Side of Happiness? How, When, and Why Happiness Is Not Always Good. ''Perspectives on Psychological Science, 6''(3), 222–233. https://doi.org/10.1177/1745691611406927 Gross, J. J. (2015). Emotion regulation: Current status and future prospects. ''Psychological Inquiry, 26''(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Gruber, J., Villanueva, C., Burr, E., Purcell, J. R., & Karoly, H. (2020). Understanding and Taking Stock of Positive Emotion Disturbance. ''Social and personality psychology compass, 14''(1), e12515. https://doi.org/10.1111/spc3.12515 Goagoses, N., Bolz, T., Eilts, J., Schipper, N., Schütz, J., Rademacher, A., Vesterling, C., & Koglin, U. (2022). Parenting dimensions/styles and emotion dysregulation in childhood and adolescence: a systematic review and Meta-analysis. ''Current Psychology, 42'', 18798–18822. https://doi.org/10.1007/s12144-022-03037-7 Hechtman, L. A., Raila, H., Chiao, J. Y., & Gruber, J. (2013). Positive Emotion Regulation and Psychopathology: A Transdiagnostic Cultural Neuroscience Approach. ''Journal of Experimental Psychopathology, 4''(5), 502–528. https://doi.org/10.5127/jep.030412 Huwaë, S., & Schaafsma, J. (2016). Cross‐cultural differences in emotion suppression in everyday interactions. ''International Journal of Psychology, 53''(3), 176–183. https://doi.org/10.1002/ijop.12283 Haag, A.-C., Bagrodia, R., & Bonanno, G. A. (2024). Emotion Regulation Flexibility in Adolescents: A Systematic Review from Conceptualization to Methodology. ''Clinical Child and Family Psychology Review, 27''(3), 697–713. https://doi.org/10.1007/s10567-024-00483-6 Kashdan, T. B., Young, K. C., & Machell, K. A. (2015). Positive emotion regulation: addressing two myths. ''Current Opinion in Psychology, 3'', 117–121. https://doi.org/10.1016/j.copsyc.2014.12.012 Linehan, M. (1993). ''Cognitive-behavioral treatment of borderline personality disorder.'' Guilford Press Livingstone, K. M., & Srivastava, S. (2012). Up-regulating positive emotions in everyday life: Strategies, individual differences, and associations with positive emotion and well-being. ''Journal of Research in Personality, 46''(5), 504–516. https://doi.org/10.1016/j.jrp.2012.05.009 Miller-Slough, R. L., & Dunsmore, J. C. (2023). Parents” and friends” socialization of positive emotions: Associations with adolescent emotion regulation. ''Journal of Applied Developmental Psychology, 88'', 101579. https://doi.org/10.1016/j.appdev.2023.101579 Mansueto, G., Palmieri, S., Salatini, L., Piccioni, S., Ruggiero, G. M., Sassaroli, S., Spada, M. M., & Caselli, G. (2025). Positive Emotion Dysregulation: A Metacognitive Perspective. ''Clinical Psychology & Psychotherapy, 32''(4). https://doi.org/10.1002/cpp.70109 Roth, L. H. O., Bencker, C., Lorenz, J., & Laireiter, A.-R. (2024). Testing the validity of the broaden-and build theory of positive emotions: a network analytic approach. ''Frontiers in Psychology, 15''. https://doi.org/10.3389/fpsyg.2024.1405272 Vogel, A. C., Brotman, M. A., Roy, A. K., & Perlman, S. B. (2023). Review: Defining positive emotion dysregulation: Integrating temperamental and clinical perspectives. ''Journal of the American Academy of Child & Adolescent Psychiatry, 62''(3), 297–305. https://doi.org/10.1016/j.jaac.2022.06.019 Weiss, N. H., Gratz, K. L., & Lavender, J. M. (2015). Factor Structure and Initial Validation of a Multidimensional Measure of Difficulties in the Regulation of Positive Emotions. ''Behavior Modification, 39''(3), 431–453. https://doi.org/10.1177/0145445514566504 Weiss, N. H., Contractor, A. A., Forkus, S. R., Goncharenko, S., & Raudales, A. M. (2020). Positive Emotion Dysregulation Among Community Individuals: The Role of Trauma Exposure and Posttraumatic Stress Disorder. ''Journal of Traumatic Stress, 33''(5), 741–749. https://doi.org/10.1002/jts.22497 Weiss, N. H., Schick, M. R., Raudales, A. M., Forkus, S. R., Thomas, E. D., Contractor, A. A., & Sullivan, T. P. (2023). Experimental Investigation of the Influence of Positive Emotion Dysregulation on Risky Behavior Following Idiographic Emotion Inductions. ''Clinical Psychological Science, 11''(3), 490–508. https://doi.org/10.1177/21677026221133295 }} ==External links== * [https://www.psychologytoday.com/au/blog/everyday-resilience/202404/emotional-well-being-5-healthy-practices-for-regulation Emotional wellbeing] (Psychology today) * [https://www.belonghealth.com.au/emotional-regulation-dysregulation-perth-mandurah.html Emotion dysregulation consequences] (Belong health) * [https://www.londonpsychiatry.clinic/blog/5-evidence-based-emotional-regulation-strategies Emotional regulation strategies] (London psychiatry clinic) * [https://www.youtube.com/watch?v=MyfzIQH6YKI Positive emotions with Barbara Fredrickson] (Youtube.com) __TOC__ [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotion regulation]] [[Category:Motivation and emotion/Book/Positive emotion]] 0qg2wod0f0nqrapac4a5fzvcsr6hnhd 0 331038 2844015 2839404 2026-11-02T20:25:00Z Jtneill 10242 Copyediting 2844015 wikitext text/x-wiki {{title|Technology-based pain management:<br>How can technology-based tools alter pain perception and pain management?}} __TOC__ ==Overview == {{RoundBoxTop|theme=2}} [[File:Headache touching forehead.jpg|thumb|326x326px|'''Figure 1'''. A woman holding her head in discomfort, illustrating the stress that accompanies neuropathic pain, similar to Elana.]] '''Case study: Elena's shift in pain perception''' Elena is a 35-year-old graphic designer living with chronic neuropathic pain following a viral infection. For years, her pain felt unpredictable and overwhelming; some days she could work normally, while other days even holding a pencil felt impossible. What frightened her most wasn’t the pain itself, but the random timing and the belief that she had no control over it. Her pain specialist introduced her to a set of perception-altering technologies: a VR {{explain}} program designed to redirect attention and reduce pain intensity through immersive environments, a wearable sensor that tracked muscle tension and autonomic arousal, and a mobile app that visualised her pain patterns using predictive coding algorithms {{g}} Within two weeks, Elena noticed something she didn’t expect: her pain felt less threatening. The VR sessions helped her shift attention away from the pain, reducing the “alarm” response in her brain. The wearable showed spikes in tension before flare ups, helping her intervene early. The app revealed consistent patterns {{g}} stress, poor sleep, and long work sessions transforming what once felt random into something understandable and predictable. As Elena’s understanding changed, her perception changed. Pain episodes felt less catastrophic, shorter, and easier to manage. Technology didn’t eliminate her pain it altered the way her brain interpreted it, giving her a sense of agency she hadn’t felt in years. {{RoundBoxBottom}} Technology-based pain interventions are receiving increasing research attention across healthcare settings, including postoperative care, rehabilitation, and labour wards (Musters et al., 2023; Yang et al., 2023). The pain technology market is projected to increase by '''USD $70.2 billion''' over the 10 years (Statifacts.com)see annual [https://www.statifacts.com/outlook/chronic-pain-treatment-devices-market report] {{ic|Move embedded links to References and provide APA style citation}}. Virtual reality can redirect attention away from pain, TENS {{ic|explain abbreviatino}} devices can modulate pain-signal transmission, and biofeedback systems help individuals regulate physiological arousal. Virtual reality can redirect attention away from pain, TENS devices can modulate pain-signal transmission, and biofeedback systems can help individuals regulate physiological arousal. Wearables, mobile applications, and telehealth platforms are also becoming more widely used, giving people accessible ways to track pain patterns and engage in self-management (Green & Chakravarthy, 2025; Li et al., 2024). Technology has therefore become increasingly influential in how pain is monitored, interpreted and managed across healthcare. For people with chronic pain, catastrophic thoughts and fear of further pain or injury may lead to avoidance of movement and meaningful activities, contributing to reduced functioning and increased distress (Gatchel, 2016; Vlaeyen & Linton, 2000). This chapter explores how technology-based interventions may interrupt this cycle by influencing pain signals, attention, emotions, thoughts, motivation and perceived control. {{RoundBoxTop|theme=2}} '''Focus questions''' {{ic|Use bullet-points as shown in Tutorial 2}} •How does technology alter the way people perceive and respond to pain? •How do psychological processes such as attention, emotion and cognitive appraisal contribute to technology-based pain management? •How do different technologies influence people's sense of control, engagement and motivation when managing pain? {{RoundBoxBottom}} == Understanding pain perception and management with technology == [[Pain]] is not determined solely by physical injury. It is a personal sensory and emotional experience influenced by biological, psychological and social factors (Raja et al., 2020). Understanding the distinction between pain and nociception {{explain}} helps explain why technology can change a person’s pain experience even when it does not remove the original injury or health condition. Pain perception is shaped by how the brain interprets nociceptive signals alongside attention, emotions, memories, beliefs, expectations and the surrounding context (Bushnell et al., 2013; Tracey & Mantyh, 2007). Technology can alter this interpretation by redirecting attention, regulating physiological arousal, modifying pain-signal transmission or changing catastrophic thoughts about pain. It may also influence how a person responds by encouraging movement, coping and active self-management rather than fear and avoidance. Technology can therefore reduce the intensity, unpleasantness or perceived threat of pain while supporting greater confidence and control over pain-management behaviour. Nociception is the nervous system's detection system of potentially harmful events, such as heat or pressure. Pain is the sensory and emotional experience that follows. The two are closely related but not the same; for example, detecting danger does not always produce conscious pain, and pain can occur without damage (Raja et al., 2009). === Physiology of pain and pain perception === Nociceptive information acts as a pain message to the brain's "pain central pathways", called ascending pathways. One important pathway is the Spinothalamic tract. It carries the pain messages to the Thalamus, the part of the brain that helps further send information to the other parts of the brain (Basbaum et al.,2009). The Brain then works out various parts of the pain experience. Areas of the somatosensory system aid the brain to identify where the pain is located and its intensity. Other areas, including the Insula and anterior cingulate cortex, help the brain recognise how the feels and how the body should respond {{f}}. The messages can also travel in the opposite direction down descending pathways the signals act to either increase or reduce pain {{f}}. This process is called Descending modulation {{f}}. This process is assistance {{sp}} to help us explain why attention, expectations and emotions can change how much pain an individual experiences (Tracey & Mantyh, 2007). It also explains how Pain is not controlled by one singular "pain centre". Instead, several areas on the brain work together to create the pain experience (Bushnell et al.,2013). === '''Connecting the pathways to technology''' === These pathways help explain how technologies can influence different parts of the pain experience. TENS stimulates sensory nerves and may alter how nociceptive information is processed in the spinal cord, linking electrical stimulation with transmission and modulation (Moayedi & Davis, 2013). VR engages attention in an immersive environment, changing how much attention is available for pain during the experience (Malloy & Milling, 2010). Biofeedback makes selected bodily responses visible so users can practise self-regulation. Experimental research has examined whether this practice can influence both perceived pain and spinal nociceptive responses, although genuine feedback did not produce greater benefits than the comparison conditions (Arsenault et al., 2013). Connected stimulation systems can combine treatment with recorded feedback and remote monitoring to help guide treatment adjustments (Green & Chakravarthy, 2025). Monitoring therefore supports management decisions, but collecting information alone does not establish that pain processing has changed. Digital psychological programs address thoughts, emotions and coping behaviour. Their effects may appear in emotional wellbeing even when daily interference from pain does not improve. For example, the EPIO app trial found {{how}} {{who}} {{where}} improvements in depression and self-regulatory fatigue, but no significant improvement in pain interference at three months (Bostrøm et al., 2023). Across these approaches, pain intensity, unpleasantness, emotional distress and everyday functioning should be considered separately. == How does technology alter paig perception? == This section introduces various technologies that assist in the treatment of pain. Throughout the explanation of each tool, there is further suggestive evidence of how the technology ultimately alters pain perception, evidenced by Various Psychological theories. === VR and attention modulation === [[Virtual reality]] (VR) immerses users in a computer-generated environment through visual, auditory and sometimes interactive experiences {{f}}. These engaging activities can redirect attention towards the virtual environment, leaving less attention resources available for pain processing. The whole idea associates {{aw}} with limited capacity theory, when a person has limited capacity to attend the competing information, therefore an absorbing experience may reduce attention towards the pain experience. VR may therefore reduce pain intensity or unpleasantness during use. {{g}} (Li et al., 2011). To provide a practical example of VR in pain reduction, a study by Musters et al. 2023) {{g}} found that twenty-four women {{where}} in induced labour trialed a VR-guided meditation and interactive game. Women experienced {{g}} 26% pain decrease, finding that the women benefited in particular from meditation and described the VR as providing distraction from the intensity of labour pain. ===TENS=== Transcutaneous electrical nerve stimulation (TENS) delivers electrical impulses via electrodes placed on the skin to stimulate sensory nerves (Blincoe, 2007). The effects are associated with gate control theory {{f}}, which proposes that it can actively reduce pain signals sent to the spinal cord. This is commonly described as "closing the gate", reducing the information being sent to the brain and potentially reducing the intensity of pain (Moayedi& Davis, 2013). (TENS) {{g}} may also stimulate the release of the brain's pain-relieving substances such as endorphins and enkephalins (Blincoe,{{g}}2007). An example of (TENS) {{g}} is the NXSTIM Eco AI program described by (Green & Chakravarthy,2025). This system combines (TENS) {{g}} with electrical muscle stimulation, a smartphone interface and remote monitoring. Treatment response information helps guide adjustments to stimulation settings, supporting individualised treatment plans {{f}}. This illustrates how electrical stimulation can be combined with a feedback system to adjust treatment. === Biofeedback and autonomic regulation === Biofeedback uses sensors to measure bodily responses and display the information; users then use the recorded information to support regulation. This is associated with self-regulation theory {{f}}. For example, (Lazaridou et al. 2023) {{g}} studied {{where}} people with chronic lower back pain in an 8-week program. The participants used feedback about their muscle activity to learn how to control muscle tension. Those receiving feedback were guided through relaxation to visualise the effectiveness. Participants receiving biofeedback reported {{ic|how much?}} lower intensity of pain. Further studies suggest that practising self-regulation via biofeedback can influence the pain experience, although biofeedback is used alongside therapeutic guidance {{f}}. Overall, biofeedback provides information about physiological pain responses, enabling users to practice and develop greater control over their pain (Arsenault,2013). === Cognitive Digital CBT and ACT tools === CBT and ACT {{ic|explain abbreviations}} are digital programs which aim to help people manage the thoughts, feelings, and behaviours that shape their pain experience. Apps and online activities allow users to practice coping skills increase engagement and improve functioning even though the pain is still an ongoing occurrence. these {{g}} tools relate to both cognitive appraisal, self-regulation theory and the fear avoidance model. According to (Gatchel etal.,2007) {{g}} these tools address the psychological and behavioural influences on pain alongside psychical and social factors. An example of these tools is EPIO {{ic|explain abbreviation}} {{g}} an app-based program focusing on CBT and ACT after {{missing}} 3-month period participants reported {{ic|how much?}} improvements related to depression and fatigue related to pain management. In another {{g}} example {{who}} participants using EPIO showed {{ic|how much?}} greater reductions in catastrophic thinking than those receiving normal care, these included less dwelling on pain and less exaggeration of threating pain (Solberg Nes et al., 2024). These demonstrate a benefit to thinking patterns that make pain overwhelming. === Wearable sensors and Mobile apps === Wearable sensors are used to help develop pain management plans by recording treatment use with a participant's pain ratings. This allows and clinicians to follow changes over time and make informed treatment adjustments. This allows users to track themselves overtime and make informed treatment adjustments. An example of this (Green & Chakravarthy, 2025) {{g}} Describe NXTIM EcoAI, as a wearable system that deliver electrical stimulation and connects via a smartphone app. Users rate their pain before and after treatment, while the device automatically records the information such as treatment duration and stimulation setting. The system uses feedback to recommend adjustments suited to the individual. Users and clinician can also view their progress remotely. Thes approach supports self-regulation theory through monitoring feedback and adjustments (Karoly,1993) {{ic|Are there more recent citations?}}. === Telehealth and remote monitoring === Telehealth and remote platforms assist individuals to receive care at home and in rural areas where they can continue care. Remote contact via telehealth can improve pain perception by making these individuals more confident in their ability to manage pain whilst remaining actively supported. (Green & Chakravarthy 2025) {{g}} describe how platforms for home-based treatment make pain more manageable by giving the person a clearer understanding of what they can do when their symptoms change. This relates to self-efficacy, the belief that one can self-manage challenges (Bandura,1997). Recognition of progress and receiving practical encouragement may strengthen this belief, shifting the pain experience from overwhelming to manageable. {{g}} (Jackson et al., 2014) found {{how}} greater self-efficacy was associated with lower perceived pain, less stress and better functioning. Together, this suggests telehealth could influence pain perception through supported confidence and a greater sense of control. == Psychological theories behind technology‑based pain management == This section explains the psychological theories associated with technology-based pain management, and how perception of the "pain experience" is altered and shaped in various ways. * '''Biopsychosocial model of pain:''' The biopsychosocial model explains that pain arises from interactions between biological processes, psychological factors and social influences. Biological influences include nervous system activity; psychological influences include thoughts, emotions and expectations. For example, {{g}} (Williams & Craig 2016) explain that understanding pain requires recognition of its cognitive and social dimensions alongside sensation and emotion. When applied to technology, this model explains the value of combining physical treatment with psychological supports and healthcare professionals. Addressing these influences can assist with making the pain experience more manageable. * '''Limited‑capacity theory of attention:''' The limited-capacity theory explains that if attention is limited, then pain competes with other resources. Research by (Buhle & Wager ,2010) {{g}} demonstrates this relationship through experimentation, finding that completing a task that directs attention elsewhere can reduce pain intensity even if the pain stimulation continues. This explains why changes in attention alter the pain experience. The psychological benefit is that perceptual attention to pain is altered. * '''Gate control theory of pain:''' Gate control theory, introduced by (Melzack & Wall 1965) {{g}}, explains that the spinal cord acts like a gate regulating how much information about harmful stimuli reaches the brain. They further propose that activity in the sensory nerves, along with signals travelling down from the brain, can open or close this gate. When the gate closes, less information passes through, reducing the intensity of pain. This helps explain why non-painful sensations can help reduce pain; for example, when an individual hurts themselves, they rub or hold where it hurts, therefore reducing the pain. * '''Self‑regulation theory:''' Self‑regulation theory explains how people guide their thoughts, emotions and behaviour towards a goal. In pain management, this involves setting achievable goals, reviewing progress and adjusting actions. (Hamilton et al.,2004) {{g}} explains that pain and negative emotions can disrupt this process by drawing attention towards discomfort and narrowing the goals a person pursues. Self-regulation helps people respond purposefully to pain, such as balancing activity and rest while continuing towards goals. This supports a sense of control and stops pain from dominating everyday decisions. * '''Operant conditioning:''' Operant conditioning explains how the consequences of an action influence whether it is repeated. Application is explained by (Fordyce et al.,1973) {{g}} in regard to chronic pain, explaining how responses others and relief from discomfort can reinforce pain-related behaviours. For example, avoiding an activity may bring immediate relief from anxiety. {{g}} making avoidance more likely next time. Encouragement and recognition of achievable goals can instead reinforce participation. This explains how consistent positive consequences can strengthen helpful pain-management behaviours and support everyday functioning. This further supports technology-based interventions. * '''The Transactional theory of stress and coping:''' This theory is explained by (Lazarus & Folkman,1984) {{g}} as judging the situation as threatening and assessing our ability to cope, which shapes emotions and behaviour. Applied to pain, this means the experience is shaped by both how threatening the pain seems and how capable the person feels about managing it. Interpretation of pain as overwhelming while believing there is little one can do increases distress. Recognising available coping strategies can make the pain more manageable. (Turk & Wilson 2010) explain catastrophic interpretations of pain increase fear, focus attention and encourage avoidance. {{ic|How is this related to technology?}} * '''Psychological flexibility model:''' The Psychological flexibility model explains how individuals can respond to difficult thoughts, feelings and sensations whilst acting in ways that reflect their values. (Hayes et al., 2012) explains how processes including acceptance, present- moment awareness and recognising thoughts rather than facts. Applied to pain, this means that negative thoughts do not have to determine actions. Acceptance reduces the struggle with discomfort, while values motivate participation in meaningful activities. Supporting this approach (Veerhof et al.,2016) found that acceptance -and mindfulness interventions reduced pain interference and anxiety across 25 randomised trials. The model therefore explains how changing a person's relationship to pain can make it less dominant in everyday life and support continued participation in valued activities. {{ic|How is this related to technology?}} * '''Social cognitive theory and self‑efficacy:''' Social cognitive theory explains how beliefs, behaviour and the social environment influence one another. A central concept is self-efficacy, a person's belief in their ability to carry out actions and manage challenges. (Bandura, 1997) explains that this confidence develops through successful experience, observing others, encouragement and interpretations of bodily and emotional responses. Applied to pain, self-efficacy means believing that one can manage discomfort and continue meaningful activities. This belief supports motivation, effort and persistence when pain presents as a challenge. (Jackson et al., 2014) analysed 86 samples involving 15,616 people, found that greater self-efficacy was associated with lower pain intensity, less emotional distress and less functional impairment. These findings support the importance of perceived capability in both the pain experience and everyday functioning. {{ic|How is this related to technology?}} * '''Affective‑motivational modulation of pain:''' Affective-motivational modulation describes how emotions and personal goals shape the Pain experience and our responses. "Affective" refers to feelings and emotional distress, while "motivational" refers to the drive to act, such as seeking relief, protecting the body or pursuit of meaningful goals or activities. (Mordernall et al.,2026) {{g}} Explain that emotional and cognitive contexts can change pain perception through several patterns of brain activity. These include descending regulation, where the brain influences the processing of incoming nociceptive information. This helps explain how a change in emotional experience can alter pain even when the stimulus remains the same. (Vervoort & Trost ,2018) extend this perspective into a social context, explaining how emotions and competing goals shape interactions between those experiencing pain and those supporting them. For example, seeking protection and reassurance may compete with a desire for independence or participation {{g}}. Applied to technology-based pain management, this perspective suggests that tools supporting emotional regulation and meaningful goals may influence both the "pain experience " and how people respond to it. * '''Fear‑avoidance model of pain:''' The fear-avoidance model explains how interpreting pain as threatening can influence behaviour. (Vlaeyen & Linton., 2000) {{g}} describe how catastrophic interpretation, such as believing that the movement will cause further injury, can produce fear and encourage avoidance. A person may also become increasingly alert to bodily sensations, monitoring them for signs of danger. Avoiding a feared activity can bring relief from anxiety, strengthening the tendency to avoid it again. However, avoidance limits opportunities to discover that an activity may be manageable and does not necessarily produce a feared outcome. Gradually confronting appropriate activities offers an opportunity to revise these expectations and reduce fear ( Vlaeyen &Linton 2000). The model therefore provides a foundation for understanding the relationship between pain-related beliefs, fear and behaviour , explored further in the following section on Chronic pain and avoidance. {{ic|How is this related to technology?}} === Chronic pain and avoidance === [[File:Pain cycle.jpg|thumb|326x326px|'''Figure 2'''. The pain avoidance cycle [https://pmc.ncbi.nlm.nih.gov/articles/PMC3820295/ cycle]]] * Chronic pain refers to pain that persists or recurs for longer than three months. Chronic pain may affect attention, memory, mood, sleep, motivation, relationships, and participation in daily activities (Gatchel, 2016). Avoidance can lead to reduced activity, physical fitness and confidence. If avoidance persists, it can lead to long-term impacts such as limiting functional capacity and causing long-term disability. {{ic|How is this related to technology?}} * Catastrophising involves a form of catastrophic thinking where pain is interpreted as a fearful experience leading to fear of movement or reinjury. Pain Catastrophising is known to be both psychologically and physically damaging if not addressed. This is explained by the fear- avoidance model. Technology-based interventions may target different parts of the pain avoidance cycle. {{ic|How is this related to technology?}} * Digital CBT targets catastrophic thinking by challenging the thoughts in the self -help program. An example of the program would include recording harmful thoughts, evidence or challenging the belief to develop a balanced thought. (Bostrom et al.,2023). Digital ACT is a guided online Self-help program targeting the link between fear and avoidance; an example of the program would include noticing harmful thoughts, stepping back, allowing feelings, identifying the value and choosing a more meaningful action. By targeting '''avoidance behaviours''' through guided online programs, delivered technologies can alter perception of pain as a barrier to activity, helping people take manageable steps towards more meaningful activities despite discomfort ( Trompetter et al.,2015). * '''See Figure 2''' One The Avoidance Pain Cyle == Visualising the brain's pain pathways == [[File:Pain pathway diagram showing transduction, transmission, modulation, and perception.png|thumb|293x293px|Figure 2: The Brain's Pain Pathways ]] *'''The figure shown as Figure 2:''' The Brain's Pain Pathways' Shows how the pain experience occurs. Potentially harmful stimuli are converted into nerve Signals '''Transduction''' carried through '''Transmission''' and processed into '''Perception''' experience, increasing or reducing during '''modulation''' ' (Tracey & Mantyh,2007). See the following examples: Perception via immersive experiences redirects attention, which can reduce intensity during use (Malloy & Milling, 2010). TENS - Transmission and modulation Via electrical stimulation works by activating sensory nerves and can reduce the onward transmission of nociceptive information to the spinal cord (Moayedi & Davis, 2013) Wearables, Gamification, Telehealth and digital psychological programs can support management and coping throughout the whole pain experience. == Future Applications and Emerging Technologies == Imagine a pain-management device that disappears when its work is done, or a virtual limb that moves in response to muscle signals. Researchers are exploring technologies that could change how we monitor, experience, and treat pain. * '''Disappearing nerve implants:''' A soft implant could wrap around a nerve, cool it to interrupt pain signals, and then gradually dissolve into the body. (Reeder et al.,2022) demonstrates the approach using rats with neuropathic pain . {{g}} * '''Brain Implants:''' A device could soon recognise patterns associated with a person's pain ( Shrivalker etal.,2023) {{g}} used implanted electrodes and machine learning to predict changes in reported chronic pain. This study could provide foundations for future systems that recognise pain and automatically adjust stimulation. * '''Virtual Limbs:''' Extended reality can recreate a missing limb on screen. Sensors record activity in remaining muscles, and machine learning translates these signals into movements of the virtual limb. This allows people with phantom limb pain to practice controlling a digital version of their missing limb (Lendaro et al,2025 ). * '''Sound Powered Brain Stimulation:''' An implanted device could receive its power through ultrasound rather than a battery. (Zhang et al.,2022) developed a device that converts ultrasound energy into electricity to stimulate a brain region involved in pain modulation. Experiments in rats demonstrated pain-relieving effects, opening possibilities for future wireless implants powered from outside the body. == Limitations and ethical considerations == Technology-based pain management offers new and exciting possibilities, but their value depends on evidence, accessibility affordability and responsible use. Tools should support individual needs while protecting safety, privacy and choice. * '''Effectiveness of results may vary:''' Benefits differ between individuals, pain condtions and technologies. Immediate relief does not necessarily mean lasting effects or long-term improvements to functioning . {{g}} {{f}} * '''Limited evidence on emerging technologies:''' Some futuristic technologies have only been tested in small studies or animal research. Their long-term effectiveness and safety require further investigation (Demarest et al.,2024) . * '''Cost and accessibility:''' Device costs, internet access and digital literacy can limit participation and cause inequalities in care (Li et al., 2024). * '''User Engagement:''' Regular practice and ongoing support may be needed. Difficulties using technologies or maintaining engagement can limit usefulness (Li et al., 2024). * '''Side effects and safety:''' VR can cause dizziness, fatigue, eye strain and neck discomfort. Risks should be assessed for each technology and user (U.S FDA, n.d) * '''Privacy and data security:''' Pain apps, wearables and brain monitoring systems collect sensitive information. Users should know what is collected, who has access and how their information is protected. * '''Informed consent:''' Users need clear explanations of the benefits, risks, alternatives and experimental status so they can decide participation. (WHO,2021) * '''AI bias and accountability:''' Biased training data can produce unequal or inaccurate recommendation. AI systems evaluation, transparency and human oversight (WHO,2021) * '''Respect for lived experiences:''' Device readings should support assessment without dismissing the person reported pain, which must be respected (Raja et al.,2020) {{ic|Content past this point is ignored for marking purposes due to being over the maximum word count}} '''Table 1''' See The Summary Table below : How pain-management technologies work and the theories that help explain them {| class="wikitable" style="margin: auto; |- ! Technology !! How it alters pain perception !! Theory / Model |- | VR || Redirects attention || Limited-capacity attention |- | TENS || Modulates pain signals || Gate control theory |- |Biofeedback |Support regulation of arousal |Self-regulation, conditioning |- |CBT/ACT tools |Changes to coping, appraisal and avoidance |Fear avoidance, flexibility |- |Wearables/apps |Records patterns and provides feedback |Self-regulation |- |Telehealth |Remote support, health access and treatment |Self-efficacy |} ;Quizzes Check your knowledge below by trying the interactive Quiz :<quiz display="simple"> {Virtual reality (VR) can reduce pain by redirecting a person’s attention away from painful stimuli?: |type="()"} + True - False {TENS devices work by increasing the intensity of pain signals sent to the brain. Answer?: |type="()"} - True + False </quiz> ==Conclusion== Technology-based tools can alter pain perception and management by influencing how pain signals are processed, where attention is directed, how bodily and emotional responses are regulated and what pain means to the individual experiencing it. VR, TENS, biofeedback and digital psychological programs act through different mechanisms, but share an important foundation: that is, pain is a personal experience shaped by biological, psychological and social influences. The psychological theories explain how these tools can support changes beyond immediate relief. Redirecting attention may make pain less intrusive, changing threating interpretations may reduce distress and practicing coping skills may strengthen confidence and perceived control. Feedback, achievable goals and supportive contact can also encourage motivation and continued participation in meaningful activities. Effective pain management therefore includes improvements in emotional wellbeing, everyday functioning and the ability to pursue valuable goals, even when pain persists. The practical message is to match technology to the person's needs, preferences and goals within their individual plan. Its effectiveness relies on research, participation, accessibility and safety whilst acknowledging personal experience. As emerging technologies expand what is possible, these principles remain essential for turning innovative tools into mutually beneficial applications for use. As our case study explains, Elena's changes assisted her in making her "pain experience " less threatening and more manageable . Technology assisted Elena to recognise her patterns, practice coping and regain her sense of agency. This captures the central idea of technology-based pain management , to help people gain greater control of their pain and better quality of life . {{tip|Focus Questions Answered: How does technology alter the way people perceive and respond to pain? Technology can influence pain signals, redirect attention and support changes in thoughts, emotions and behaviour. VR provides distraction, TENS modulates pain-signal transmission, and biofeedback helps users practice regulating bodily responses. Digital psychological tools support coping and participation in everyday activities, even when pain persists. How do attention, emotion and cognitive appraisal contribute to technology-based pain management? Attention influences how strongly pain occupies awareness, emotions shape its unpleasantness, and cognitive appraisal affects how threatening or manageable it seems. VR draws attention towards engaging experiences, biofeedback supports regulation of arousal, and digital CBT helps users reconsider catastrophic thoughts. ACT encourages acceptance and meaningful action despite discomfort. How do different technologies influence control, engagement and motivation? Wearables and apps provide feedback that helps people recognise patterns, monitor progress and adjust their actions. Biofeedback supports active practice, while telehealth provides guidance and encouragement. Achievable goals and greater confidence in coping can strengthen motivation to continue managing pain and participating in valued activities. }} ==See also== [[wikipedia:Virtual_reality_therapy|Virtual reality therapy]] (Wikipedia) [[wikipedia:Transcutaneous_electrical_nerve_stimulation|Transcutaneous electrical nerve stimulation]] (Wikipedia) * https://en.wikipedia.org/wiki/Pain {{ic|Create internal link}} * https://en.wikipedia.org/wiki/Virtual_reality {{ic|Create internal link}} * [[Motivation and emotion/Book/2021/Self-regulation|Self-regulation]] (Book chapter, 2021) * [[wikipedia:Transcutaneous_electrical_nerve_stimulation|Transcutaneous electrical nerve stimulation]] (Wikipedia) * [[wikipedia:Cognitive_behavioral_therapy|Cognitive behavioral therapy]] (Wikipedia) * [[wikipedia:Biopsychosocial_model|Biopsychosocial model]] (Wikipedia) * [[wikipedia:Gate_control_theory|Gate control theory]] (Wikipedia) {{tip|Why do we rub a sore spot?... 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C. de C., & Craig, K. D. (2016). Updating the definition of pain. Pain, 157(11), 2420–2423. https://doi.org/10.1097/j.pain.0000000000000613 39. World Health Organization. (2021). Ethics and governance of artificial intelligence for health: WHO guidance. https://www.who.int/publications/i/item/9789240029200 40. Yang, X., Zhong, S., Yang, S., He, M., Xu, X., He, S., Fan, G., & Liu, L. (2023). Global scientific trends in virtual reality for pain treatment from 2000 to 2022: Bibliometric analysis. JMIR Serious Games, 11, Article e48354. https://doi.org/10.2196/48354 41. Zhang, T., Liang, H., Wang, Z., Qiu, C., Peng, Y. B., Zhu, X., Li, J., Ge, X., Xu, J., Huang, X., Tong, J., Ou-Yang, J., Yang, X., Li, F., & Zhu, B. (2022). Piezoelectric ultrasound energy-harvesting device for deep brain stimulation and analgesia applications. Science Advances, 8(15), Article eabk0159. https://doi.org/10.1126/sciadv.abk0159 }} * {{tip|*Key take aways: * Pain is more than a physical signal. Attention, emotions, beliefs and social influences shape the experience. * Different technologies work in different ways. VR redirects attention, TENS modulates pain signals, and biofeedback supports bodily regulation. * Thoughts and coping matter. Digital CBT and ACT tools can help address catastrophic thinking and support meaningful activity. * Feedback and support can build confidence. Wearables, apps and telehealth help people monitor patterns and take an active role in managing pain. * Progress is about living better. Greater control, less distress and participation in valued activities matter—even when pain persists. }} ==External links== * https://www.iasp-pain.org/resources/terminology/ * https://www.qld.gov.au/health/services/specialists/persistent-pain/understanding-pain/managing-your-pain * https://aci.health.nsw.gov.au/chronic-pain/painbytes/pain-and-mind-body-connection {{tip|Take the next step: Learning about pain is a starting point. Explore the resources suggested to better understand your pain, discover coping strategies and consider which tools might suit your needs. Small, manageable steps can help you put that knowledge into practice. }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Pain]] [[Category:Motivation and emotion/Book/Technology]] t8u3aea96hee3z8861a50kjqismrjmm 0 331099 2844073 2839578 2026-11-03T00:56:19Z Jtneill 10242 Copyediting 2844073 wikitext text/x-wiki {{title|Emotion dysregulation:<br>What is emotion dysregulation, what are its consequences, and how can it be managed?}} __TOC__ ==Overview== {{RoundBoxTop|theme=2}} [[File:Burnout At Work - Occupational Burnout.jpg|Burnout_At_Work_-_Occupational_Burnout|220px|right|thumb|'''Figure 1.''' Taylor feels frustrated after the submissions portal froze at a critical moment.]] '''Scenario''' Taylor is trying to upload an assignment last minute but the submissions portal freezes exactly as he presses “submit”. He watches the cursor’s loading icon spin again and again, likely because the portal is overwhelmed by other last minute submissions. Though {{g}} there is still 15 minutes before the deadline, Taylor can feel his heart rate increase, jaw tighten and leg shake. Overwhelmed, Taylor finds himself aggressively clicking the laptop trackpad and becoming consumed by a feeling of frustration. Then the trackpad becomes unresponsive. Taylor shoves his laptop aside and drops his head onto the table, angrily questioning why nothing ever seems to go his way (see Figure 1). {{RoundBoxBottom}} From moments of [[wikipedia:Happiness|happiness]] and excitement to [[wikipedia:Sadness|sadness]] or irritation, [[w:emotions|emotions]] are an integral aspect of being human, and it is expected that these emotions fluctuate as we navigate the complexity of our everyday lives. Emotions are neither ‘good’ nor ‘bad’, rather emotions have a specific [[wikipedia:Evolution|evolutionary]] purpose that has helped us survive, adapt and interact with others (Šimić et al., 2021). Nonetheless, there are times when strong and overwhelming emotions can impair our daily functioning. This is called [[wikipedia:Emotional_dysregulation|emotion dysregulation]]. Emotion dysregulation can be defined as the difficulty to [[wikipedia:Coping|cope]] with intense emotions, to the extent that an individual's [[wikipedia:Emotional_responsivity|emotional response]] is disproportionate to the situation and they have difficulty implementing adaptive [[wikipedia:Coping|coping strategies]] (Gross & Thompson, 2007, as cited in Aslan et al., 2024). This chapter investigates the processes underlying emotion dysregulation, its behavioural and psychological consequences, and practical approaches to facilitate [[w:Emotional_self-regulation|emotion regulation]]. Through understanding these ideas, individuals can be better prepared to identify and respond to emotion dysregulation in [[wikipedia:Adaptive_behavior|adaptive ways]]. {{RoundBoxTop|theme=2}} '''Focus questions''' * What is emotion dysregulation? * How does emotion dysregulation impact daily functioning and psychological well-being? * What approaches can help manage emotion dysregulation? {{RoundBoxBottom}} ==What is emotion dysregulation? == Emotions can be understood as functional [[wikipedia:Motivation|motivational forces]] that enable people to meet their [[Motivation and emotion/Book/2013/Needs|needs]] and accomplish their [[wikipedia:Goal|goals]] (Fischer & Manstead, 2016). Brief in duration, emotions arise rapidly to motivate action and then dissipate, preparing individuals to respond to the next meaningful event (Fischer & Manstead, 2016; Levenson, 1994). However, emotions become unhelpful when their intensity, duration or form does not match environmental demands (Gross, 2015b). For instance, [[wikipedia:Embarrassment|embarrassment]] that leads to social avoidance, [[wikipedia:Anger|anger]] that causes harm, or [[wikipedia:Guilt_(emotion)|guilt]] that drives excessive [[wikipedia:Self-blame_(psychology)|self-blame]] (Gross, 2015b).   Emotion dysregulation can therefore be conceptualised as “a pattern of emotional experience and/or expression that interferes with appropriate goal-directed behavior” (Beauchaine, 2015, p. 876). In other words, emotion dysregulation is the inability to understand and cope with the onset, intensity and expression of emotional states (Grecucci et al., 2020). Emotion dysregulation may emerge as reduced emotional awareness, intense experience and expression of emotion, emotional [[wikipedia:Rigidity_(psychology)|rigidity]] (i.e., inability to adapt feelings or reactions), and difficulty reinterpreting distressing situations (D’Agostino et al., 2017). This extends to dysregulation of both unpleasant and pleasant emotions (see [[Motivation and emotion/Book/2026/Positive emotion dysregulation|positive emotion dysregulation]]). Such patterns indicate that emotion dysregulation may arise from disruptions in [[wikipedia:Emotional_self-regulation|emotion regulation]] processes, a premise underscoring major [[wikipedia:Theory|theoretical]] models seeking to explain this construct. === Process model of emotion regulation === [[File:Gross (2007) Process Model of Emotion Regulation.png|thumb|412x412px|'''Figure 2.''' Process model of emotion regulation (Gross, 2015b).]] The [[wikipedia:Emotional_self-regulation#Process_model|process model of emotion regulation]] (Gross, 1998, as cited in Gross, 2015b) holds that emotions emerge through a predictable situation-attention-appraisal-response sequence: an event occurs, [[wikipedia:Attention|attention]] is directed toward that event, meaning is assigned and individuals respond accordingly. Moreover, this model identifies five distinct [[wikipedia:Emotional_self-regulation#Strategies|emotion regulation strategies]] individuals draw upon in response to situational demands at different points in this sequence (see Figure 2), each leading to different consequences for how people feel, think and behave (Gross, 2015b). [[wikipedia:James_Gross|Gross]] (2015b) describes each of these regulation strategies as follows: * '''Situation selection:''' refers to deliberate attempts to seek or avoid situations expected to evoke specific emotions. * '''Situation modification:''' efforts to directly modify aspects of the physical environment to alter its emotional impact. * '''Attentional deployment:''' intentionally redirects focus toward or away from aspects of a given situation. * '''Reappraisal:''' reinterpretation of one’s evaluation (i.e., thoughts) of a situation to change its emotional impact. * '''Suppression:''' attempts to prevent experiential, behavioural, or physiological reactions to an emotion after it has emerged. Thus, these regulation strategies enable individuals to [[wikipedia:Consciousness|consciously]] exert a degree of control over their emotional experience, either down-regulating unpleasant emotions or up-regulating pleasant ones. Therefore, emotion dysregulation constitutes the inability to effectively implement, execute or adapt at least one of these regulation strategies leading to maladaptive, inflexible emotional responses (Aldo et al., 2010). Interestingly, a [[wikipedia:Systematic_review|systematic review]] and [[wikipedia:Meta-analysis|meta-analysis] {{if|of how many studies}} conducted by Boemo et al. (2022), identified that suppression is associated with increased intensity and/or duration of negative emotions, and notably these effects persisted. Insofar as the authors contend that it may be considered a maladaptive regulation strategy {{g}}. Although the study centred on emotion regulation, it points to a possible relationship between suppression and emotion dysregulation, warranting further consideration. === Extended process model of emotion regulation === Gross (2015a) introduced the [[Motivation and emotion/Book/2026/Extended process model of emotion regulation|extended process model of emotion regulation (EPM)]], elaborating on the earlier framework to consider the processes that initiate emotion regulation, guide strategy selection and determine why some successfully regulate emotion while others find it difficult. In doing so, the EPM further emphasises that emotions arise from cognitive evaluations and distinguishes three stages of emotion regulation (Grecucci et al., 2020). These stages are identification (i.e., the decision whether to regulate emotion), selection (i.e., choosing a broad strategy from the original model) and implementation (i.e., the execution of a situationally appropriate specific tactic) (Gross, 2015a). A {{ic|type of?}}review conducted by Sheppes et al. (2015) surmised that emotion dysregulation occurs when individuals experience difficulty enacting any of these stages, leading to [[wikipedia:Coping#Negative_techniques_(maladaptive_coping_or_non-coping)|maladaptive coping]]. '''Identification-stage failures''' denote difficulties understanding emotional cues, such that individuals either do not initiate regulation strategies or engage in regulation efforts unnecessarily (Sheppes et al., 2015). Further, an individual may not assign value to the goal of regulating emotion or may believe that emotions are unalterable, meaning that regulation is not adequately initiated (Gross, 2015b). '''Selection-stage failures''' underscore difficulties selecting appropriate strategies (Sheppes et al., 2025). This may be attributed to a perceived lack of access to suitable strategies, either because an individual has few strategies to draw upon or excessively relies on a specific strategy (Gross, 2015b). If an individual believes themself incapable of carrying out a strategy effectively, they may not engage with it altogether (Gross, 2015b). This points to [[wikipedia:Self-efficacy|self-efficacy]] as a potential factor in determining whether individuals successfully regulate their emotions (Gross, 2015b). '''Implementation-stage failures''' mark the impaired ability to translate broad strategies into concrete tactics that are situationally appropriate, either due to lack of cognitive resources or knowledge (Sheppes et al., 2025; Gross, 2015b). In other words, an individual may identify an appropriate regulation strategy but are unable to carry out the specific tactic (e.g., reframing or breathing techniques). {{RoundBoxTop|theme=2}} '''Revisiting Taylor's scenario: ''' Despite recognising physiological indicators of his rising frustration, Taylor did not initiate an emotion regulation strategy to help manage his frustration. Instead, Taylor becomes overwhelmed by his emotions, and his ensuing impulsive actions only lead him to become further away from his goal (submitting the assignment). This demonstrates selection-stage failure causing emotion dysregulation. {{RoundBoxBottom}} {{robelbox|theme=3|title=Let's do a quick knowledge check!|icon=Paomedia small-n-flat light-bulb.svg|iconwidth=68px}}<div style="{{Robelbox/pad}}"> <quiz display=simple header=none> {A person becomes anxious before their class presentation and decides to attempt a breathing exercise. However, they don’t recall the correct method, leading them to breathe quickly and irregularly, inadvertently worsening their panic. This is an example of: | type="(+)"} - Identification-stage failure - Selection-stage failure + Implementation-stage failure </quiz> </div> {{Robelbox/close}} ==Impact of emotion dysregulation on daily functioning and psychological well-being== Emotion regulation is understood to be essential for normative development and adaptive functioning across psychological, behavioural and social domains (Weiss et al., 2015). Emotion dysregulation, in turn, undermines functioning and is often attributed to the emergence of [[wikipedia:Psychopathology|psychopathology]] and maladaptive behaviours, including [[wikipedia:Impulsivity|impulsivity]], [[wikipedia:Disinhibition|disinhibition]] and risk-taking (Weiss et al., 2015). Further, emerging as either hyporeactivity or hyperreactivity, emotion dysregulation may lead to prolonged periods of distress in which negative [[wikipedia:Affect_(psychology)|affect]] is sustained and/or positive affect is reduced (Boemo et al., 2022). This section explores the role of emotion dysregulation in [[wikipedia:Interpersonal_relationship|interpersonal relationships]], risk-taking behaviour and [[wikipedia:Mental_health|mental health]]. === Interpersonal consequences of emotion dysregulation === [[File:Empathy tiles.jpg|thumb|243x243px|'''Figure 3.''' Empathy fosters positive social interactions. ]] Humans are inherently social beings, and emotions play a foundational role in shaping social interactions through signalling [[wikipedia:Intention|intentions,]] coordinating behaviour and facilitating connection (Van Kleef, 2009). As such, social interactions necessitate the ability to recognise, understand and [[wikipedia:Empathy|empathise]] with others’ emotions (see Figure 3; Salazar Kämpf et al., 2023). Eisenberg (2000, as cited in Salazar Kämpf et al., 2023) suggests that individuals can attend to others’ emotional needs only when they are first able to attend to their own, thus depending on the ability to regulate emotions. Salazar Kämpf et al.'s (2023) meta-analysis {{ic|of how many studies?}} investigated the relationship between emotion regulation abilities, social [[wikipedia:Affect_(psychology)|affect]] and [[wikipedia:Cognition|cognition]]. They found that emotion dysregulation was [[wikipedia:Negative_relationship|negatively correlated]] {{ic|how strongly?}} with [[wikipedia:Empathy#Cognitive_empathy|cognitive empathy]], such that individuals who reported greater difficulty regulating emotions were less able to engage with [[wikipedia:Perspective-taking|perspective-taking]] (i.e., putting themselves in another’s shoes). This association may be explained by the fact that emotion dysregulation drains cognitive reserves, leaving less capacity to effectively engage in other resource-intensive cognitions, like cognitive empathy (Salazar Kämpf et al., 2023). This has significant implications for social interactions, as these findings suggest that individuals experiencing emotion dysregulation may either fail to recognise others’ needs or struggle to recognise the impact of their behaviour on those around them, leading to conflict. Interestingly, Salazar Kämpf et al. (2023) also observed a positive association between emotion dysregulation and [[wikipedia:Empathic_concern|empathic distress]], suggesting difficulties regulating emotion may impair one’s capacity to reduce the emotional impact of seeing others in distress. In other words, it implies that someone prone to emotion dysregulation is more likely to become dysregulated when confronted with another person’s distress. === Role of emotion dysregulation in risk-taking behaviours === Risk-taking behaviours can be defined as “any behaviour that puts one at risk for a negative outcome, be it physical, emotional, social or finical” (Weiss et al., 2015, p. 22). Emerging research indicates that emotion dysregulation may increase the likelihood of risky behaviours (e.g., [[wikipedia:Substance_abuse|substance abuse]] and [[wikipedia:Risky_sexual_behavior|risky sexual behaviour]]), as individuals either attempt to alleviate aversive emotional states or seek pleasant emotional states (Weiss et al., 2015). Therefore, one could argue that risk-taking functions as a suppression-based regulation strategy. However, while risky behaviours may offer short-term benefits, they often give rise to unintended paradoxical outcomes in the long-term (Weiss et al., 2015). In 2023, Li et al. investigated {{how}} whether emotion dysregulation [[wikipedia:Mediation_(statistics)|mediated]] the role between self-reported [[wikipedia:Sensation_seeking|sensation seeking]] (i.e., tendency to engage high-arousal experiences) and dangerous driving. The study found that elevated sensation seeking was correlated with greater emotion regulation difficulties, and this relationship predicted increased risky driving behaviour. Further, Hayley et al. (2017) found that reduced levels of emotional awareness and the inability to control negative emotions increased the likelihood of making risky or ill-conceived decisions when driving. Thus, implying that emotion dysregulation may reduce an individual’s capacity to recognise and control risky behaviour {{g}}. === Emotion dysregulation as a transdiagnostic risk factor === Alongside its impact on everyday functioning, research has indicated that frequent experiences of emotion dysregulation can negatively affect our [[wikipedia:Well-being|well-being]] and may be a transdiagnostic risk factor for the development and maintenance of mental health disorders (Beauchaine & Cicchetti, 2019; Aslan et al., 2024). Transdiagnostic factors refer to underlying [[wikipedia:Mental_disorder#Risk_factors|risk]], maintenance or [[wikipedia:Protective_factor|protective factors]] across a diverse range [[wikipedia:Mental_disorder|mental health disorders]], transcending diagnostic criteria (Dalgeish et al., 2020). Beauchaine and Cicchetti (2019) observed that emotion dysregulation has been associated with [[wikipedia:Internalizing_disorder|internalizing disorders]], [[wikipedia:Externalizing_disorder|externalizing disorders]], [[wikipedia:Personality_disorder|personality disorders]] and psychotic disorders. In particular, emotion dysregulation in children and adolescents may be a predisposing factor to the emergence of mental health disorders in adulthood (Cole et al., 2017). For example, difficulties regulating emotions are a hallmark feature of [[wikipedia:Post-traumatic_stress_disorder|post-traumatic distress disorder]] (PTSD), often leading to periods of severe emotional distress (Agathos et al., 2025). In a prospective study, Pencea et al. (2020) identified that emotion dysregulation during a traumatic event significantly predicted the persistence of PTSD symptoms, even after controlling for other risk factors. Indicating that immediate [[wikipedia:Psychotherapy|psychotherapeutic]] interventions targeting emotion regulation may reduce the development of PTSD. Further, a systematic review {{ic|of how many studies?}} conducted by Agathos et al. (2025) identified distinct stage-specific patterns of emotion dysregulation in PTSD across the EPM framework (see Table 1). Identification-stage findings found that emotional clarity was negatively correlated with PTSD severity during period of negative affect. Further, individuals with PTSD demonstrated a tendency to rely on suppression regulation strategies (e.g., avoidance) when confronted with [[wikipedia:Stressor|stressors]]; however, these strategies were associated with greater symptom severity, suggesting failures at the selection-stage of regulation. Taken together, these studies indicate that emotion dysregulation not only increases [[wikipedia:Vulnerability|vulnerability]] to developing PTSD but continues as a symptom that contributes to the ongoing maintenance of the disorder. '''Table 1''' ''Difficulties in Implementing Emotion Regulation Strategies and PTSD.'' {| class="wikitable" !Stage in emotion regulation cycle !Deficit in PTSD contributing to emotion dysregulation |- |Identification-stage |Reduced emotional awareness and clarity |- |Selection-stage |Overeliance on maladaptive strategies (suppression) |} == How can emotion dysregulation be managed? == [[File:Practicing mindfulness promotes creativity.png|thumb|190x190px|'''Figure 4.''' Emotion regulation skills promote general well-being.]] Research in [[wikipedia:Developmental_psychology|developmental psychology]] suggests that emotion regulation capabilities are not innate, rather regulation strategies develop through [[wikipedia:Learning|learning]] processes across the lifespan (Wright et al., 2025). As emotion dysregulation fundamentally encompasses the difficulty implementing adaptive regulation strategies, the fact that these skills can be improved is promising for the management of emotion dysregulation and promotion of general well-being (see Figure 4).   === Cognitive reappraisal strategies for dysregulation === [[Motivation and emotion/Book/2025/Cognitive strategies and emotion regulation|Cognitive strategies]] are among the most empirically supported tools to promote adaptive emotion regulation, particularly among those with mental health disorders (Kazantzis et al., 2018). Specifically, research suggests that the frequent use of reappraisal is associated with greater positive affect, reduced negative affect and enhanced interpersonal functioning (Brockman et al., 2017; Boemo et al., 2022). Cognitive reappraisal involves intentionally reinterpreting the meaning of emotion-eliciting stimuli in a way that alters one’s emotional response, enabling the shift from negative affect toward more neutral emotional states (Wang & Yin, 2023). This strategy is typically beneficial to guide individuals in recognising, and amending, maladaptive thoughts (Wang & Yin, 2023). In clinical settings, cognitive reappraisal forms a core component of [[wikipedia:Cognitive_behavioral_therapy|cognitive behavioural therapy]], effectively reducing emotional distress (Wang & Yin, 2023; Kazantzis et al., 2018).   Nonetheless, the effectiveness of cognitive reappraisal remains situationally dependent (Wang & Yin, 2023), not unlike other emotion regulation strategies. Implementing this strategy universally could lead to paradoxical effects on emotion dysregulation (Wang & Yin, 2023). For instance, reappraisal may be adaptive when a stressor is uncontrollable, yet counterproductive when the stressor is controllable and one could directly modify the situation (Troy et al., 2013, as cited in Brockman et al., 2017). === Mindfulness strategies for dysregulation === [[Motivation and emotion/Book/2013/Mindfulness|Mindfulness]] has emerged as an alternative to traditional response-focused emotion regulation strategies (Brockman et al., 2017). Specifically, mindfulness may facilitate adaptive engagement with, and expression of, emotions, mitigating the likelihood of hypo- or hyperreactivity (Brockman et al., 2017). While cognitive reappraisal in concerned with altering the negative content of maladaptive thoughts, mindfulness seeks to regulate emotions through awareness and acceptance (Chambers et al., 2009, as cited in Brockman et al., 2017). Mindfulness may be practiced regularly, or implemented to reduce emotion dysregulation in the moment, involving a range of practices, including structured breathing exercises and [[wikipedia:Meditation|meditation]] (Guendelman et al., 2017). Roemer et al.’s (2017) {{ic|what type of?}} review highlights that mindfulness may reduce emotion dysregulation through strengthening one’s ability to recognise physiological cues of emotion and redirect attention away from perceived threats. Thus, suggesting mindfulness functions to enhance one’s capacity to detect when regulation is necessary and engage in adaptive strategies before emotional responses escalate. In other words, mindfulness may help mitigate identification-stage failures as described by Gross (2015b). Notably, however, a significant limitation of research in this area concerns the [[wikipedia:Operationalization|operationalisation]] of mindfulness as an emotion regulation strategy (Guendelman et al., 2017). The breadth of techniques and psychological phenomena labelled as mindfulness, along with varying views on whether mindfulness is a practice or mental state, denote the complexity of this construct (Guendelman et al., 2017). Future research should seek to identify specific mindfulness-based techniques that most effectively manage emotion dysregulation, both over time and in the moment. ==Conclusion== Emotions are a core component of the human experience, and while emotions are neither ‘good’ nor bad’, there are times when they become overwhelming and dominate our behaviours. Emotion dysregulation encompasses the moments when emotions become maladaptive, as their intensity, duration and/or expression exceeds what the situation actually demands, regardless of whether the emotions are pleasant or unpleasant. According to the process model of emotion regulation framework (Gross, 2015a, 2015b), impairments in the identification, selection and implementation of regulation strategies underscore the emergence of emotion regulation difficulties. Emotion dysregulation has been linked to interpersonal difficulties, impaired decision-making (i.e., increased risk-taking) and emergence of mental health disorders, indicating that emotion dysregulation has consequences across domains of functioning. Nonetheless, emotion regulation skills, in part, can be learned. Individuals can learn to manage intense emotions with conscious thinking and enhanced recognition of physiological cues, through practicing the use of cognitive reappraisal and mindfulness.    {{RoundBoxTop|theme=3}} '''Take-home messages: ''' Emotion dysregulation refers to persistent difficulties managing intense emotional states. Consequently, individuals may struggle with interpersonal relationships and decision-making (e.g., risk taking) when experiencing heightened emotions. Strengthening cognitive reappraisal and mindfulness skills can support the management of emotion dysregulation. {{RoundBoxBottom}} ==See also== * [[Motivation and emotion/Book/2024/ADHD and emotional regulation|ADHD and emotional regulation]] (Book chapter, 2024) * [[Motivation and emotion/Book/2025/Cognitive strategies and emotion regulation|Cognitive strategies and emotion regulation]] (Book chapter, 2025) * [[w:Emotional_self-regulation|Emotional self-regulation]] (Wikipedia) * [[Motivation and emotion/Book/2026/Extended process model of emotion regulation|Extended process model of emotion regulation]] (Book chapter, 2026) ==References== {{Hanging indent|1= Agathos, J., Yurtbasi, M., O'Brien, H., & Putica, A. (2025). 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Emotion dysregulation and emerging psychopathology: A transdiagnostic, transdisciplinary perspective. {{em|Development and Psychopathology, 31}}, 799–804. https://doi.org/10.1017/S0954579419000671 Boemo, T., Nieto, I., Vazquez, C., & Sanchez-Lopez, A. (2022). Relations between emotion regulation strategies and affect in daily life: A systematic review and meta-analysis of studies using ecological momentary assessments. {{em|Neuroscience and Biobehavioral Reviews, 139}}, Article 104747. https://doi.org/10.1016/j.neubiorev.2022.104747 Brockman, R. Ciarrochi, J., Parker, P., & Kashdan, T. (2017). Emotion regulation strategies in daily life: Mindfulness, cognitive reappraisal and emotion suppression. {{em|Cognitive Behaviour Therapy, 46}}(2). https://doi.org/10.1080/16506073.2016.1218926 Cole, P. M., Hall, S. E., & Hajal, N. J. (2017). Emotion dysregulation as a vulnerability to psychopathology. In T. P. Beauchaine and S. P. 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(2025). Learning emotion regulation: An integrative framework. {{em|The Psychological Review, 132}}(1), 172–203. https://doi.org/10.1037/rev0000506 }} ==External links== * [https://www.healthline.com/health/how-to-control-your-emotions How to become the boss of your emotions] (Healthline) * [https://www.ted.com/talks/ted_ed_how_to_manage_your_emotions How to manage your emotions] (TED-Ed) * [https://www.youtube.com/watch?v=SWvHZkkrAjc How to master your emotions & never get angry or bothered by anyone] (The Mel Robbins Podcast) * [https://www.psychologytoday.com/au/blog/click-here-for-happiness/202108/what-is-emotional-dysregulation What is emotional dysregulation?] (Psychology Today) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional self-regulation]] 2ipkq43klie6zat1fnzcnx6u44xtfn5 0 331101 2844037 2841178 2026-11-02T22:13:15Z Jtneill 10242 Copyediting 2844037 wikitext text/x-wiki {{title|Fear extinction:<br>What psychological and neural processes underlie the extinction of fear responses?}} __TOC__ ==Overview == {{RoundBoxTop|theme=3}} [[File:Caliber 3 protection dog.jpg|Caliber_3_protection_dog|thumb|150px|'''Figure 1'''. A dog attacking a person, reinforcing fear conditioning which can help develop a fear response whenever the affected person sees a dog {{g}}. Imagine you're bitten by a dog (see Figure 1). You might develop a fear response whenever you see a dog because the brain has learned: Dog → Danger → Fear Subsequently, if you consistently have positive encounters with dogs without any negative consequences, you will gradually learn: Dog → No Danger → Safety What are the psychological and neurological mechanisms involved? {{RoundBoxBottom}} '''Fear extinction''' is a decline in conditioned fear responses (CRs) following nonreinforced exposure to a feared conditioned stimulus (CS) (Myers, 2006). Extinction is not simply the erasure of a fear memory, it involves learning a new association that inhibits the original fear response. If an individual experiences positive outcomes while facing the fearful stimulus without receiving any maladaptive consequences you are gradually learning a process called extinction learning {{g}}. Cognitively, extinction occurs when the expected threatening outcome does not occur. Repeated safe experiences weaken the expression of fear responses and strengthen the safety association (Maren et al., 2013). However, the original fear can remain. This is reiterated by renewal, where fear returns in different circumstances, reinstatement, where fear reemerges after experiencing the threat again, and spontaneous recovery, where fear returns after a period of time (Quirk & Mueller, 2007). Several brain regions interact during fear extinction. The '''amygdala''' is essential in the attainment and expression of fear. The '''ventromedial prefrontal cortex''' (vmPFC) contributes to regulation of fear responses and the retrieval of extinction memories (Milad & Quirk, 2002). The '''hippocampus''' supplies contextual information, thus, helping decide if a previous threat should be perceived as dangerous or safe (Maren et al., 2013). Therefore, fear extinction involves new inhibitory learning rather than the erasure of the original fear memory. The interaction between hippocampus, vmPFC, and amygdala provides individuals {{missing}} the ability to learn that prior threatening stimuli can be safe, resulting in a reduction of perceived fear responses when danger is not displayed. {{RoundBoxTop|theme=1}} '''Focus questions''' * How does fear extinction change the way individuals respond to previously threatening stimuli? * What psychological processes contribute to the development of new safety associations during fear extinction? * How do the amygdala, vmPFC, and hippocampus interact to regulate fear extinction? * Why might fear responses return after successful extinction, and what does this reveal about how fear memories are stored and regulated? {{RoundBoxBottom}} == Understanding fear extinction == [[Motivation and emotion/Book/2022/Fear|Fear]] is an intervening response that assists individuals in detecting and responding to context-dependent stimuli (Adolphs, 2013). Through the use of [[wikipedia:Fear_conditioning|fear conditioning]], neutral stimuli can have a maladaptive correlation with perceived danger, {{g}} this results in a learned fear response. It is important to note that when the expected harm has failed to occur repeatedly, individuals can determine that the stimulus is not dangerous, {{g}} this is a process identified as fear extinction (Furini et al., 2014). However, extinction is not regarded as the deletion of the original fear; it is simply the formation of new safe inhibitory associations that aids in the reduction of fear. Extinction learning is influenced by contextual information, this helps identify if a past threatening stimuli should be perceived as dangerous or safe (Bouton, 2004; Quirk & Mueller, 2007; Maren et al., 2013). Having a greater understanding of these processes grants a foundation for analysing psychological and neurological mechanisms that allows fear responses to be reduced as well as explaining why fear has the potential of returning after successful extinction. === Formation of fear and fear conditioning === *'''Fear conditioning''' occurs during previous neutral stimuli becoming associated with a threatening event, consequently causing the stimulus to later trigger a fear response. This learned connection is essential for understanding how fear can be ultimately reduced through extinction (Quirk & Mueller, 2007). *When a fear has been conditioned, individuals can acquire fearful responses via maladaptive stimuli that may mirror the original danger. We can use the dog example again where an individual bitten by a dog can potentially gain a fear of other dogs, even when there is no clear danger. This is known as '''Generalisation of fear''' (Dymond et al., 2015). *'''Individual differences in fear learning''' proposes that prior experience, perceived vulnerability, and learning history can influence the intensity of the developed associated fear. These distinctions can affect how individuals respond to harmful stimuli and determine how easily they learn that the stimuli are safe. {{f}} === Fear extinction as new learning === *'''Fear Extinction''' is a result of previously threatening stimuli being repeatedly experienced without the expected threat. Instead of erasing the original fear memory, extinction allows learning new inhibitory safety associations, resulting in the individual to distinguish that the stimuli is no longer a predictive danger (Quirk & Mueller, 2007). *'''Inhibitory Associations''' occur during extinction when people learn that prior dangerous stimuli can no longer accurately predict an aversive outcome. This allows new learning to support the inhibiting of expression of the original fear response without fully erasing the original association (Bouton, 2004). *'''Repeated exposure and expectation''': Consistently facing fearful stimuli without experiencing the predicted danger can help individuals update their expectations. This process is identified as expectancy violation which is significant in inhibitory learning approaches to reducing fear (M. G. Craske et al., 2008). === The role of context and memory in extinction === *Extinction learning is influenced by the context in which it occurs. The brain uses situational information to determine if a prior threatening stimuli should be perceived as unsafe or secure. This assists in clarifying why fear can return when the stimulus has appeared during a variety of scenarios (Bouton, 2004). *Extinction memories are potentially easier to retrieve in the environment where the extinction occurred. When individuals face their previously frightening stimuli in a new environment, the original fear response may reemerge (Bouton, 2004). *Due to fear extinction not necessarily erasing the original fearful memory, fear can return following a delay in time or exposure to a threatening event. This signifies that extinction incorporates learning to regulate fear instead of simply forgetting it (Bouton, 2004). == Psychological processes underlying fear extinction == Psychological processes underlying fear extinction proposes how individuals acquire the ability to reduce fear responses when prior maladaptive stimuli lose the capability to forecast threats. Instead of forgetting fearful experiences, '''inhibitory learning''' provides new adaptive associations to rival the original fear memory (M. G. Craske et al., 2008). An integral psychological process is expectancy violation, occurring when expected danger fails to develop, thus, encouraging individuals to revise their assumptions regarding danger. It is important to note that extinction learning is influenced via situational factors, therefore, expression of fear can differ regarding the context that the stimuli is encountered in. These mechanisms demonstrate how fear responses have the ability to reduce while keeping the original fearful memory. Thus, having a greater comprehensive insight in inhibitory learning, prediction error, and contextual memory retrieval reveals how fear extinction occurs and why relapse occurs after extinction is achieved. === Inhibitory learning and the formation of safety associations === *Fear extinction is acknowledged {{f}} as a process of new inhibitory learning, instead of the complete deletion of the original fear memory. As previously stated, when a past dangerous stimuli is consistently presented without the expected maladaptive outcome, the individual begins to associate that the stimuli is safe. Let's use the dog scenario again; for example, someone who is afraid of dogs after being bitten may slowly learn through positive encounters that dogs do not necessarily anticipate danger. This new association inhibits the expression of the original fear response, allowing for adaptive behaviour regardless of the persistence of the original fear memory (Quirk & Mueller, 2007; Furini et al., 2014). *Repeated encounters with previous harmful stimuli without negative consequences can potentially reinforce safety learning. Over a period of time, individuals can become more confident that the stimulus does not always predict danger, thus reducing fear responses in common scenarios (M. G. Craske et al., 2008). *Inhibitory learning supports individuals to develop more flexible versatile responses by learning that stimuli previously associated with danger can occur without a maladaptive response. Although, this safety association may rival rather than permanently replace the original fear memory (Bouton, 2004). === Prediction error and expectancy violation === * [[wikipedia:Expectancy_violations_theory|Expectancy violation]] is an integral psychological mechanism of extinction. This occurs when the outcome an individual expects differs from what actually happens. During the occurrence of extinction, an individual may predict a maladaptive outcome but consistently experience its absence, {{g}} this is known as [https://pmc.ncbi.nlm.nih.gov/articles/PMC3518876/ prediction error]. This mismatch causes prediction error, notifying that previous expectations about threats need to be updated. Therefore, greater expectancy violation can increase extinction learning by motivating the individual to reconsider their beliefs between the relationship of the stimulus and threat (M. Craske, 2015). * '''Prediction error''' utilises information that an individual's existing expectations has the potential to contain inaccuracies. When the predicted threat consistently fails to occur, this mismatch can encourage the individual to revise their understanding of the stimulus and develop a greater association between the stimulus and safety (M. Craske, 2015) * '''Exposure experiences''' can provide more information when outcomes may differ from the expected outcome. For example, an individual expecting a dog to attack may learn that their prediction was wrong after a successful encounter, thus, having a positive correlation in the reduction of dangerous expectations during future encounters (M. G. Craske et al., 2008). === Context, memory retrieval and the return of fear === * Context and memory retrieval is another process that influences extinction learning significantly {{f}}. The learned safety association during extinction does not automatically replace the original fear association; rather, the brain must retrieve the appropriate memory depending on the context of the situation. Ultimately, fear has the potential to return through processes such as [https://www.alleydog.com/glossary/definition.php?term=Renewal+Effect renewal], when the stimulus is engaged in different contexts, or [https://www.sciencedirect.com/topics/computer-science/spontaneous-recovery spontaneous recovery], when adequate time has passed. This reinforces that successful extinction depends on both the safety of the stimulus and the ability to retrieve that safety learning when it is required (Maren et al., 2013). *'''Fear renewal''' is prominent when an extinguished fear response reemerges when an individual faces the conditioned stimuli in a different context rather than the extinction environment. This highlights that extinction memories might be context-dependent instead of being universally retrieved (Bouton, 2004). *'''Spontaneous recovery''' highlights the reemergences of prior reduced fear response following a period of time. This cognitive process implies that the original fear memory can remain after extinction, thus, highlighting the significance of maintaining and retrieving safety learning over time (Bouton, 2004). == Neural mechanisms of fear extinction == [[File:PTSD stress brain.gif|thumb|Figure 2. Regions of the limbic system and prefrontal cortex - key components for fear and memory processing.|171x171px]] Neural mechanisms of fear extinction highlights the neurological interactions amongst regions of the brain which are responsible for cognitive processes like; fear learning, emotional responses, memory retrieval and contextual processing. Despite extinction learning reducing maladaptive responses {{g}} it does not always delete the original fearful stimuli, rather, neural activity alters communication between brain regions supporting the regulation and expression of fear and the retrieval of freshly acquired extinction memories (Quirk & Mueller, 2007). There are three essential neurological structures utilised during fear extinction (see in figure 2 and 3); the amygdala, the ventromedial prefrontal cortex (vmPFC), and the hippocampus. Exploring the functions of these regions will provide a greater understanding regarding how these neural mechanisms are responsible for the acquisition and expression of fear and why previous fearful responses can reemerge. === The amygdala: acquisition and expression of fear === * The amygdala plays an integral role in the acquisition and expression of conditioned fear. During fear conditioning, the [https://pmc.ncbi.nlm.nih.gov/articles/PMC10604397/ basolateral amygdala] helps establish associations between previous neutral stimulus and an aversive outcome, additionally, the [[wikipedia:Central_nucleus_of_the_amygdala|central amygdala]] contributes to the expression of physiological and behavioural fear responses {{f}}. When fear extinction is occurring, the amygdala remains involved due to the original fear association staying. However, extinction-related neural changes allow for the reduction of fear expression when the stimulus is no longer linked to danger (Quirk & Mueller, 2007). *The amygdala contains a plethora of interconnected nuclei inside the medial temporal lobe which is associated in processing emotional stimuli. The basolateral and central regions aid in identifying threat association and planning defensive approaches (LeDoux, 2000). *Fear conditioning contains alterations in synaptic communication within the amygdala, specifically via glutamate-mediated plasticity {{f}}. When extinction occurs, GABAergic circuits assist in regulating fear-related neural activity, thus, reducing conditioned fear responses (Myers, 2006). === The vmPFC: inhibition and retrieval of extinction memories === * The [[wikipedia:Ventromedial_prefrontal_cortex|ventromedial prefrontal cortex (vmPFC)]] is vital for regulating fear following extinction learning {{f}}. Instead of removing the original fear association, the vmPFC contributes to the retrieval of newly learned safety associations and can employ inhibitory influence over amygdala-based fear responses. Literature by Quirk and Mueller (2007) identified {{how}} that activity within the medial prefrontal cortex has a positive correlation with successful extinction memory retrieval, signifying the importance in suppressing prior conditioned fear. * Neuroimaging conducted on individuals has demonstrated a positive correlation in reinforced vmPFC activity during extinction-memory retrieval with reduced conditioned fear responses {{f}}. Therefore, highlighting a positive correlation with vmPFC's involvement in recalling past learned safety information (Milad et al., 2007) {{g}}. * The '''vmPFC''' contributes to the regulation of fear via interactions with the amygdala circuits. {{what}} Evidence by Milad and Quirk (2002) implies that medial prefrontal activity can coordinate extinction-memory expression, thus, helping suppress defensive responses when previously harmful stimuli are no longer predicted to be perceived as dangerous. === The hippocampus: contextual regulation of fear extinction === [[File:Gehirn_Frontalschnitt_hippocampus.png|alt=Figure 3. The Hippocampus amongst many other regions in the brain.|thumb|218x218px|Figure 3. The Hippocampus amongst many other regions in the brain.]] * The [https://my.clevelandclinic.org/health/body/hippocampus Hippocampus] delivers significant contextual information crucial for identifying if a previously threatening stimulus should be interpreted as dangerous or safe. It has a positive relationship with the vmPFC due to its classification in which extinction learning occurred and assists in guiding retrieval of the appropriate memory. This contextual processing further explains why fear can return when a previously feared stimulus is encountered outside the extinction context, demonstrating that extinction is highly dependent on context-sensitive memory retrieval (Maren et al., 2013). * The '''hippocampus''' assists in forming representations of the environment in which fear conditioning and extinction occur. Contextual memories support distinguishing scenarios that are associated with threat from those associated with safety (Maren et al., 2013). *The hippocampus cooperates with prefrontal and amygdala networks to influence which fear-related memories are gathered. This interaction reinforces why individuals may demonstrate reduced fear in familiar environments but renewed fear in unfamiliar environments (Maren et al., 2013). '''Table 1. Brain regions neural mechanisms and their relation to fear extinction''' {{f}} {| class="wikitable" |+ !Brain Region / Mechanism !Role In Fear Extinction !Neural Mechanism |- !Amygdala |Involved in the acquisition and expression of conditioned fear, while also participating in extinction learning. |During extinction, changes within the amygdala support the learning of new extinction associations. Extinction does not simply remove the original fear association. |- !vmPFC |Supports the inhibition and retrieval of extinction memories. |The vmPFC integrates information about the stimulus and extinction learning and can exert regulatory influences over amygdala activity. |- !Hippocampus |Provides contextual information that helps determine whether a previously threatening stimulus is currently safe. |The hippocampus communicates contextual information to prefrontal and amygdala circuits, helping distinguish the extinction context from contexts associated with threat. |- !Amygdala-vmPFC Interaction |Allows the brain to regulate the expression of previously learned fear. |The vmPFC can influence amygdala circuits involved in fear expression, supporting the retrieval of the safety association. |- !Hippocampus-vmPFC Interaction |Allows extinction memories to be retrieved appropriately according to context. |Contextual information from the hippocampus helps the vmPFC determine whether extinction or fear memories should guide behaviour. |} == The return of fear following extinction == * '''Renewal: The influence of Context on Fear''' *Fear can return when a previously extinguished stimulus is encountered in a context that differs from where extinction learning occurred. This phenomenon is known as '''renewal''', it demonstrates that extinction does not erase the original fear memory {{f}}. Instead, the brain retains both the original fear association and the newer safety association, with contextual information influencing which memory is expressed. The hippocampus is important in providing contextual information that helps determine whether the stimulus should be interpreted as threatening or safe (Bouton,2004; Maren et al., 2013). === Reinstatement: the return of fear following exposure to threat === * '''[https://www.sciencedirect.com/science/article/abs/pii/S0005796704002347 Reinstatement]''' occurs when a fear response returns after an individual experiences the aversive event again following extinction. For example, if someone successfully extinguishes their fear of dogs but is subsequently bitten by a dog, their fear of dogs may return. This implies that extinction learning remains vulnerable to subsequent threatening experiences, which can reactivate or strengthen the original fear association (Quirk & Mueller, 2007; Maren et al., 2013) === Spontaneous recovery and persistance of the original fear memory === *'''Spontaneous recovery''' refers to the return of a previously extinguished fear response following the passage of time. This phenomenon provides further evidence that extinction does not permanently eliminate the original fear memory. Rather, extinction creates a competing inhibitory safety memory that can weaken or suppress fear under appropriate conditions. As time passes, the original association may regain influence, demonstrating the enduring nature of fear memories and the importance of continued extinction learning (Bouton,2004; Quirk & Mueller, 2008). {{RoundBoxTop|theme=9}} ;Case study: Return of fear through context change Vansteenwegen et al. (2004) studied human participants using the fear-conditioning paradigm. Participants first learned to associate a neutral visual stimulus with an aversive loud noise. During extinction, the stimulus was repeatedly presented without the noise, causing their fear responses to decrease. However, extinction took place in a different context from where the original fear was learned. When participants were ultimately returned to the original fear-conditioning context, their conditioned fear response returned. This was identified through an increase in electrodermal responses and increased expectations that the negative outcome would occur. Participants who experienced acquisition extinction and testing in the same context did not show the same recovery of fear. This demonstrates evidence of renewal, showing that extinction does not erase the original fear memory. Instead, extinction appears to create a new safety association that is strongly influenced by context. When the individual returns to a context associated with the original threat, the original fear memory can regain control over behaviour. This finding also highlights the importance of the hippocampus and contextual processing in determining whether fear or safety memories are retrieved. {{RoundBoxBottom}} ;Test your knowledge <quiz display="simple"> {The vmPFC contributes to fear extinction by inhibiting fear responses generated by the amygdala: |type="()"} + True - False {The amygdala is primarily responsible for forming and expressing conditioned fear responses, but extinction does not completely erase the original fear memory: |type="()"} + True - False {Fear conditioning occurs when a previously neutral stimulus becomes associated with a threatening event, causing it to trigger a conditioned fear response: |type="()"} + True - False {Fear extinction permanently erases the original fear memory, preventing the fear response from returning: |type="()"} - True + False {Fear generalisation occurs when an individual develops fear responses towards stimuli that resemble the original conditioned threat: |type="()"} + True - False {Expectancy violation occurs when an anticipated threatening outcome happens exactly as the individual predicted: |type="()"} - True + False {The hippocampus contributes to fear extinction by processing contextual information that influences whether fear or extinction memories are expressed: |type="()"} + True - False {Fear renewal occurs when a previously extinguished fear response returns following a change in environmental context: |type="()"} + True - False {Spontaneous recovery occurs when fear returns because an individual experiences a new threatening event after extinction: |type="()"} - True + False {The amygdala, vmPFC and hippocampus operate independently during fear extinction, without influencing one another: |type="()"} - True + False </quiz> ==Conclusion== Fear extinction is a complex learning process through which individuals learn that a previously threatening stimulus no longer predicts danger. Rather than erasing the original memory of fear, extinction involves the development of a new inhibitory safety association that reduces the expression of fear responses (Quirk & Mueller, 2007). Psychological processes, such as expectancy violation, prediction error, and contextual memory retrieval, contribute to this new learning by allowing individuals to update their expectations about potential threats. At a neural level, interactions between the amygdala, vmPFC, and hippocampus support regulation and retrieval of extinction memories. The amygdala contributes to fear expression, while the vmPFC supports fear inhibition and the hippocampus provides contextual information to guide memory retrieval (Maren et al., 2013). However, the persistence of original fear associations means that conditioned fear can return via renewal, reinstatement, or spontaneous recovery, highlighting that extinction memories do not always override previously learned threat. Ultimately, understanding these psychological and neural mechanisms signifies how fear extinction promotes individuals to regulate conditioned fear while keeping the original fear. ==See also == * [wikipedia:Central_nucleus_of_the_amygdala][[wikipedia:Central_nucleus_of_the_amygdala|Central Amygdala]] (Wikipedia) * [[wikipedia:Expectancy_violations_theory|Expectancy Violation]] (Wikipedia) * [[Motivation and emotion/Book/2022/Fear|Fear]] (Chapter 27, 2022) * [[wikipedia:Fear_conditioning|Fear conditioning]] (Wikipedia) * [wikipedia:Ventromedial_prefrontal_cortex][[wikipedia:Ventromedial_prefrontal_cortex|Ventromedial Prefrontal Cortex (vmPFC)]] (Wikipedia) == References == {{Hanging indent|1= Adolphs, R. (2013). The biology of fear. Current Biology, 23(2), R79–R93. National Library of Medicine. https://doi.org/10.1016/j.cub.2012.11.055 Bouton, M. E. (2004). Context and behavioral processes in extinction. Learning & Memory, 11(5), 485–494. https://doi.org/10.1101/lm.78804 Craske, M. (2015). Optimizing exposure therapy for anxiety disorders: An inhibitory learning and inhibitory regulation approach. Verhaltenstherapie, 25(2), 134–143. https://doi.org/10.1159/000381574 Craske, M. G., Kircanski, K., Moriel, Z., Mystkowski, J., Chowdhury, N., & Baker, A. (2008). Optimizing inhibitory learning during exposure therapy. Behaviour Research and Therapy, 46(1), 5–27. https://doi.org/10.1016/j.brat.2007.10.003 Dymond, S., Dunsmoor, J. E., Vervliet, B., Roche, B., & Hermans, D. (2015). Fear generalization in humans: Systematic review and implications for anxiety disorder research. Behavior Therapy, 46(5), 561–582. https://doi.org/10.1016/j.beth.2014.10.001 Furini, C., Myskiw, J., & Izquierdo, I. (2014). The learning of fear extinction. Neuroscience & Biobehavioral Reviews, 47, 670–683. https://doi.org/10.1016/j.neubiorev.2014.10.016 LeDoux, J. E. (2000). Emotion circuits in the brain. Annual Review of Neuroscience, 23(1), 155–184. https://www.annualreviews.org/docserver/fulltext/neuro/23/1/annurev.neuro.23.1.155.pdf?expires=1791735526&id=id&accname=au&checksum=66010AC47123ED188645AABCFA4C7BA7 Maren, S., Phan, K. L., & Liberzon, I. (2013). The contextual brain: Implications for fear conditioning, extinction and psychopathology. Nature Reviews Neuroscience, 14(6), 417–428. https://doi.org/10.1038/nrn3492 Milad, M. R., & Quirk, G. J. (2002). Neurons in medial prefrontal cortex signal memory for fear extinction. Nature, 420(6911), 70–74. https://doi.org/10.1038/nature01138 Milad, M. R., Wright, C. I., Orr, S. P., Pitman, R. K., Quirk, G. J., & Rauch, S. L. (2007). Recall of fear extinction in humans activates the ventromedial prefrontal cortex and hippocampus in concert. Biological Psychiatry, 62(5), 446–454. https://doi.org/10.1016/j.biopsych.2006.10.011 Myers, K. M. (2006). Different mechanisms of fear extinction dependent on length of time since fear acquisition. Learning & Memory, 13(2), 216–223. https://doi.org/10.1101/lm.119806 Quirk, G. J., & Mueller, D. (2007). Neural mechanisms of extinction learning and retrieval. Neuropsychopharmacology, 33(1), 56–72. https://doi.org/10.1038/sj.npp.1301555 Vansteenwegen, D., Hermans, D., Vervliet, B., Francken, G., Beckers, T., Baeyens, F., & Eelen, P. (2004). Return of fear in a human differential conditioning paradigm caused by a return to the original acquistion context. Behaviour Research and Therapy, 43(3), 323–336. https://doi.org/10.1016/j.brat.2004.01.001 }} ==External links== * [https://my.clevelandclinic.org/health/body/hippocampus Hippocampus] (Cleveland Clinic) * [https://pmc.ncbi.nlm.nih.gov/articles/PMC3518876/ How Prediction Errors Shape Perception, Attention, and Motivation (National Library of Medicine)] * [https://www.sciencedirect.com/science/article/abs/pii/S0005796704002347 Reinstatement of Fear Responses in Human Aversive Conditioning] (ScienceDirect) * [https://www.alleydog.com/glossary/definition.php?term=Renewal+Effect Renewal Effect] - (Alleydog) * [https://www.sciencedirect.com/topics/computer-science/spontaneous-recovery Spontaneous Recovery] (ScienceDirect) * [https://pmc.ncbi.nlm.nih.gov/articles/PMC10604397/ The Basolateral Amygdala: The Core of a Network for Threat Conditioning, Extinction, and Second-Order Threat Conditioning] (National Library of Medicine) * [https://www.sciencedirect.com/science/article/abs/pii/S0149763414002681 The learning of fear extinction] (Science Direct) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Fear]] [[Category:Motivation and emotion/Book/Learning]] nzxo2o1b4hjii7mgphe73fzj812msz2 2844038 2844037 2026-11-02T22:13:42Z Jtneill 10242 2844038 wikitext text/x-wiki {{title|Fear extinction:<br>What psychological and neural processes underlie the extinction of fear responses?}} __TOC__ ==Overview == {{RoundBoxTop|theme=3}} [[File:Caliber 3 protection dog.jpg|Caliber_3_protection_dog|thumb|150px|'''Figure 1'''. A dog attacking a person, reinforcing fear conditioning which can help develop a fear response whenever the affected person sees a dog {{g}}]]. Imagine you're bitten by a dog (see Figure 1). You might develop a fear response whenever you see a dog because the brain has learned: Dog → Danger → Fear Subsequently, if you consistently have positive encounters with dogs without any negative consequences, you will gradually learn: Dog → No Danger → Safety What are the psychological and neurological mechanisms involved? {{RoundBoxBottom}} '''Fear extinction''' is a decline in conditioned fear responses (CRs) following nonreinforced exposure to a feared conditioned stimulus (CS) (Myers, 2006). Extinction is not simply the erasure of a fear memory, it involves learning a new association that inhibits the original fear response. If an individual experiences positive outcomes while facing the fearful stimulus without receiving any maladaptive consequences you are gradually learning a process called extinction learning {{g}}. Cognitively, extinction occurs when the expected threatening outcome does not occur. Repeated safe experiences weaken the expression of fear responses and strengthen the safety association (Maren et al., 2013). However, the original fear can remain. This is reiterated by renewal, where fear returns in different circumstances, reinstatement, where fear reemerges after experiencing the threat again, and spontaneous recovery, where fear returns after a period of time (Quirk & Mueller, 2007). Several brain regions interact during fear extinction. The '''amygdala''' is essential in the attainment and expression of fear. The '''ventromedial prefrontal cortex''' (vmPFC) contributes to regulation of fear responses and the retrieval of extinction memories (Milad & Quirk, 2002). The '''hippocampus''' supplies contextual information, thus, helping decide if a previous threat should be perceived as dangerous or safe (Maren et al., 2013). Therefore, fear extinction involves new inhibitory learning rather than the erasure of the original fear memory. The interaction between hippocampus, vmPFC, and amygdala provides individuals {{missing}} the ability to learn that prior threatening stimuli can be safe, resulting in a reduction of perceived fear responses when danger is not displayed. {{RoundBoxTop|theme=1}} '''Focus questions''' * How does fear extinction change the way individuals respond to previously threatening stimuli? * What psychological processes contribute to the development of new safety associations during fear extinction? * How do the amygdala, vmPFC, and hippocampus interact to regulate fear extinction? * Why might fear responses return after successful extinction, and what does this reveal about how fear memories are stored and regulated? {{RoundBoxBottom}} == Understanding fear extinction == [[Motivation and emotion/Book/2022/Fear|Fear]] is an intervening response that assists individuals in detecting and responding to context-dependent stimuli (Adolphs, 2013). Through the use of [[wikipedia:Fear_conditioning|fear conditioning]], neutral stimuli can have a maladaptive correlation with perceived danger, {{g}} this results in a learned fear response. It is important to note that when the expected harm has failed to occur repeatedly, individuals can determine that the stimulus is not dangerous, {{g}} this is a process identified as fear extinction (Furini et al., 2014). However, extinction is not regarded as the deletion of the original fear; it is simply the formation of new safe inhibitory associations that aids in the reduction of fear. Extinction learning is influenced by contextual information, this helps identify if a past threatening stimuli should be perceived as dangerous or safe (Bouton, 2004; Quirk & Mueller, 2007; Maren et al., 2013). Having a greater understanding of these processes grants a foundation for analysing psychological and neurological mechanisms that allows fear responses to be reduced as well as explaining why fear has the potential of returning after successful extinction. === Formation of fear and fear conditioning === *'''Fear conditioning''' occurs during previous neutral stimuli becoming associated with a threatening event, consequently causing the stimulus to later trigger a fear response. This learned connection is essential for understanding how fear can be ultimately reduced through extinction (Quirk & Mueller, 2007). *When a fear has been conditioned, individuals can acquire fearful responses via maladaptive stimuli that may mirror the original danger. We can use the dog example again where an individual bitten by a dog can potentially gain a fear of other dogs, even when there is no clear danger. This is known as '''Generalisation of fear''' (Dymond et al., 2015). *'''Individual differences in fear learning''' proposes that prior experience, perceived vulnerability, and learning history can influence the intensity of the developed associated fear. These distinctions can affect how individuals respond to harmful stimuli and determine how easily they learn that the stimuli are safe. {{f}} === Fear extinction as new learning === *'''Fear Extinction''' is a result of previously threatening stimuli being repeatedly experienced without the expected threat. Instead of erasing the original fear memory, extinction allows learning new inhibitory safety associations, resulting in the individual to distinguish that the stimuli is no longer a predictive danger (Quirk & Mueller, 2007). *'''Inhibitory Associations''' occur during extinction when people learn that prior dangerous stimuli can no longer accurately predict an aversive outcome. This allows new learning to support the inhibiting of expression of the original fear response without fully erasing the original association (Bouton, 2004). *'''Repeated exposure and expectation''': Consistently facing fearful stimuli without experiencing the predicted danger can help individuals update their expectations. This process is identified as expectancy violation which is significant in inhibitory learning approaches to reducing fear (M. G. Craske et al., 2008). === The role of context and memory in extinction === *Extinction learning is influenced by the context in which it occurs. The brain uses situational information to determine if a prior threatening stimuli should be perceived as unsafe or secure. This assists in clarifying why fear can return when the stimulus has appeared during a variety of scenarios (Bouton, 2004). *Extinction memories are potentially easier to retrieve in the environment where the extinction occurred. When individuals face their previously frightening stimuli in a new environment, the original fear response may reemerge (Bouton, 2004). *Due to fear extinction not necessarily erasing the original fearful memory, fear can return following a delay in time or exposure to a threatening event. This signifies that extinction incorporates learning to regulate fear instead of simply forgetting it (Bouton, 2004). == Psychological processes underlying fear extinction == Psychological processes underlying fear extinction proposes how individuals acquire the ability to reduce fear responses when prior maladaptive stimuli lose the capability to forecast threats. Instead of forgetting fearful experiences, '''inhibitory learning''' provides new adaptive associations to rival the original fear memory (M. G. Craske et al., 2008). An integral psychological process is expectancy violation, occurring when expected danger fails to develop, thus, encouraging individuals to revise their assumptions regarding danger. It is important to note that extinction learning is influenced via situational factors, therefore, expression of fear can differ regarding the context that the stimuli is encountered in. These mechanisms demonstrate how fear responses have the ability to reduce while keeping the original fearful memory. Thus, having a greater comprehensive insight in inhibitory learning, prediction error, and contextual memory retrieval reveals how fear extinction occurs and why relapse occurs after extinction is achieved. === Inhibitory learning and the formation of safety associations === *Fear extinction is acknowledged {{f}} as a process of new inhibitory learning, instead of the complete deletion of the original fear memory. As previously stated, when a past dangerous stimuli is consistently presented without the expected maladaptive outcome, the individual begins to associate that the stimuli is safe. Let's use the dog scenario again; for example, someone who is afraid of dogs after being bitten may slowly learn through positive encounters that dogs do not necessarily anticipate danger. This new association inhibits the expression of the original fear response, allowing for adaptive behaviour regardless of the persistence of the original fear memory (Quirk & Mueller, 2007; Furini et al., 2014). *Repeated encounters with previous harmful stimuli without negative consequences can potentially reinforce safety learning. Over a period of time, individuals can become more confident that the stimulus does not always predict danger, thus reducing fear responses in common scenarios (M. G. Craske et al., 2008). *Inhibitory learning supports individuals to develop more flexible versatile responses by learning that stimuli previously associated with danger can occur without a maladaptive response. Although, this safety association may rival rather than permanently replace the original fear memory (Bouton, 2004). === Prediction error and expectancy violation === * [[wikipedia:Expectancy_violations_theory|Expectancy violation]] is an integral psychological mechanism of extinction. This occurs when the outcome an individual expects differs from what actually happens. During the occurrence of extinction, an individual may predict a maladaptive outcome but consistently experience its absence, {{g}} this is known as [https://pmc.ncbi.nlm.nih.gov/articles/PMC3518876/ prediction error]. This mismatch causes prediction error, notifying that previous expectations about threats need to be updated. Therefore, greater expectancy violation can increase extinction learning by motivating the individual to reconsider their beliefs between the relationship of the stimulus and threat (M. Craske, 2015). * '''Prediction error''' utilises information that an individual's existing expectations has the potential to contain inaccuracies. When the predicted threat consistently fails to occur, this mismatch can encourage the individual to revise their understanding of the stimulus and develop a greater association between the stimulus and safety (M. Craske, 2015) * '''Exposure experiences''' can provide more information when outcomes may differ from the expected outcome. For example, an individual expecting a dog to attack may learn that their prediction was wrong after a successful encounter, thus, having a positive correlation in the reduction of dangerous expectations during future encounters (M. G. Craske et al., 2008). === Context, memory retrieval and the return of fear === * Context and memory retrieval is another process that influences extinction learning significantly {{f}}. The learned safety association during extinction does not automatically replace the original fear association; rather, the brain must retrieve the appropriate memory depending on the context of the situation. Ultimately, fear has the potential to return through processes such as [https://www.alleydog.com/glossary/definition.php?term=Renewal+Effect renewal], when the stimulus is engaged in different contexts, or [https://www.sciencedirect.com/topics/computer-science/spontaneous-recovery spontaneous recovery], when adequate time has passed. This reinforces that successful extinction depends on both the safety of the stimulus and the ability to retrieve that safety learning when it is required (Maren et al., 2013). *'''Fear renewal''' is prominent when an extinguished fear response reemerges when an individual faces the conditioned stimuli in a different context rather than the extinction environment. This highlights that extinction memories might be context-dependent instead of being universally retrieved (Bouton, 2004). *'''Spontaneous recovery''' highlights the reemergences of prior reduced fear response following a period of time. This cognitive process implies that the original fear memory can remain after extinction, thus, highlighting the significance of maintaining and retrieving safety learning over time (Bouton, 2004). == Neural mechanisms of fear extinction == [[File:PTSD stress brain.gif|thumb|Figure 2. Regions of the limbic system and prefrontal cortex - key components for fear and memory processing.|171x171px]] Neural mechanisms of fear extinction highlights the neurological interactions amongst regions of the brain which are responsible for cognitive processes like; fear learning, emotional responses, memory retrieval and contextual processing. Despite extinction learning reducing maladaptive responses {{g}} it does not always delete the original fearful stimuli, rather, neural activity alters communication between brain regions supporting the regulation and expression of fear and the retrieval of freshly acquired extinction memories (Quirk & Mueller, 2007). There are three essential neurological structures utilised during fear extinction (see in figure 2 and 3); the amygdala, the ventromedial prefrontal cortex (vmPFC), and the hippocampus. Exploring the functions of these regions will provide a greater understanding regarding how these neural mechanisms are responsible for the acquisition and expression of fear and why previous fearful responses can reemerge. === The amygdala: acquisition and expression of fear === * The amygdala plays an integral role in the acquisition and expression of conditioned fear. During fear conditioning, the [https://pmc.ncbi.nlm.nih.gov/articles/PMC10604397/ basolateral amygdala] helps establish associations between previous neutral stimulus and an aversive outcome, additionally, the [[wikipedia:Central_nucleus_of_the_amygdala|central amygdala]] contributes to the expression of physiological and behavioural fear responses {{f}}. When fear extinction is occurring, the amygdala remains involved due to the original fear association staying. However, extinction-related neural changes allow for the reduction of fear expression when the stimulus is no longer linked to danger (Quirk & Mueller, 2007). *The amygdala contains a plethora of interconnected nuclei inside the medial temporal lobe which is associated in processing emotional stimuli. The basolateral and central regions aid in identifying threat association and planning defensive approaches (LeDoux, 2000). *Fear conditioning contains alterations in synaptic communication within the amygdala, specifically via glutamate-mediated plasticity {{f}}. When extinction occurs, GABAergic circuits assist in regulating fear-related neural activity, thus, reducing conditioned fear responses (Myers, 2006). === The vmPFC: inhibition and retrieval of extinction memories === * The [[wikipedia:Ventromedial_prefrontal_cortex|ventromedial prefrontal cortex (vmPFC)]] is vital for regulating fear following extinction learning {{f}}. Instead of removing the original fear association, the vmPFC contributes to the retrieval of newly learned safety associations and can employ inhibitory influence over amygdala-based fear responses. Literature by Quirk and Mueller (2007) identified {{how}} that activity within the medial prefrontal cortex has a positive correlation with successful extinction memory retrieval, signifying the importance in suppressing prior conditioned fear. * Neuroimaging conducted on individuals has demonstrated a positive correlation in reinforced vmPFC activity during extinction-memory retrieval with reduced conditioned fear responses {{f}}. Therefore, highlighting a positive correlation with vmPFC's involvement in recalling past learned safety information (Milad et al., 2007) {{g}}. * The '''vmPFC''' contributes to the regulation of fear via interactions with the amygdala circuits. {{what}} Evidence by Milad and Quirk (2002) implies that medial prefrontal activity can coordinate extinction-memory expression, thus, helping suppress defensive responses when previously harmful stimuli are no longer predicted to be perceived as dangerous. === The hippocampus: contextual regulation of fear extinction === [[File:Gehirn_Frontalschnitt_hippocampus.png|alt=Figure 3. The Hippocampus amongst many other regions in the brain.|thumb|218x218px|Figure 3. The Hippocampus amongst many other regions in the brain.]] * The [https://my.clevelandclinic.org/health/body/hippocampus Hippocampus] delivers significant contextual information crucial for identifying if a previously threatening stimulus should be interpreted as dangerous or safe. It has a positive relationship with the vmPFC due to its classification in which extinction learning occurred and assists in guiding retrieval of the appropriate memory. This contextual processing further explains why fear can return when a previously feared stimulus is encountered outside the extinction context, demonstrating that extinction is highly dependent on context-sensitive memory retrieval (Maren et al., 2013). * The '''hippocampus''' assists in forming representations of the environment in which fear conditioning and extinction occur. Contextual memories support distinguishing scenarios that are associated with threat from those associated with safety (Maren et al., 2013). *The hippocampus cooperates with prefrontal and amygdala networks to influence which fear-related memories are gathered. This interaction reinforces why individuals may demonstrate reduced fear in familiar environments but renewed fear in unfamiliar environments (Maren et al., 2013). '''Table 1. Brain regions neural mechanisms and their relation to fear extinction''' {{f}} {| class="wikitable" |+ !Brain Region / Mechanism !Role In Fear Extinction !Neural Mechanism |- !Amygdala |Involved in the acquisition and expression of conditioned fear, while also participating in extinction learning. |During extinction, changes within the amygdala support the learning of new extinction associations. Extinction does not simply remove the original fear association. |- !vmPFC |Supports the inhibition and retrieval of extinction memories. |The vmPFC integrates information about the stimulus and extinction learning and can exert regulatory influences over amygdala activity. |- !Hippocampus |Provides contextual information that helps determine whether a previously threatening stimulus is currently safe. |The hippocampus communicates contextual information to prefrontal and amygdala circuits, helping distinguish the extinction context from contexts associated with threat. |- !Amygdala-vmPFC Interaction |Allows the brain to regulate the expression of previously learned fear. |The vmPFC can influence amygdala circuits involved in fear expression, supporting the retrieval of the safety association. |- !Hippocampus-vmPFC Interaction |Allows extinction memories to be retrieved appropriately according to context. |Contextual information from the hippocampus helps the vmPFC determine whether extinction or fear memories should guide behaviour. |} == The return of fear following extinction == * '''Renewal: The influence of Context on Fear''' *Fear can return when a previously extinguished stimulus is encountered in a context that differs from where extinction learning occurred. This phenomenon is known as '''renewal''', it demonstrates that extinction does not erase the original fear memory {{f}}. Instead, the brain retains both the original fear association and the newer safety association, with contextual information influencing which memory is expressed. The hippocampus is important in providing contextual information that helps determine whether the stimulus should be interpreted as threatening or safe (Bouton,2004; Maren et al., 2013). === Reinstatement: the return of fear following exposure to threat === * '''[https://www.sciencedirect.com/science/article/abs/pii/S0005796704002347 Reinstatement]''' occurs when a fear response returns after an individual experiences the aversive event again following extinction. For example, if someone successfully extinguishes their fear of dogs but is subsequently bitten by a dog, their fear of dogs may return. This implies that extinction learning remains vulnerable to subsequent threatening experiences, which can reactivate or strengthen the original fear association (Quirk & Mueller, 2007; Maren et al., 2013) === Spontaneous recovery and persistance of the original fear memory === *'''Spontaneous recovery''' refers to the return of a previously extinguished fear response following the passage of time. This phenomenon provides further evidence that extinction does not permanently eliminate the original fear memory. Rather, extinction creates a competing inhibitory safety memory that can weaken or suppress fear under appropriate conditions. As time passes, the original association may regain influence, demonstrating the enduring nature of fear memories and the importance of continued extinction learning (Bouton,2004; Quirk & Mueller, 2008). {{RoundBoxTop|theme=9}} ;Case study: Return of fear through context change Vansteenwegen et al. (2004) studied human participants using the fear-conditioning paradigm. Participants first learned to associate a neutral visual stimulus with an aversive loud noise. During extinction, the stimulus was repeatedly presented without the noise, causing their fear responses to decrease. However, extinction took place in a different context from where the original fear was learned. When participants were ultimately returned to the original fear-conditioning context, their conditioned fear response returned. This was identified through an increase in electrodermal responses and increased expectations that the negative outcome would occur. Participants who experienced acquisition extinction and testing in the same context did not show the same recovery of fear. This demonstrates evidence of renewal, showing that extinction does not erase the original fear memory. Instead, extinction appears to create a new safety association that is strongly influenced by context. When the individual returns to a context associated with the original threat, the original fear memory can regain control over behaviour. This finding also highlights the importance of the hippocampus and contextual processing in determining whether fear or safety memories are retrieved. {{RoundBoxBottom}} ;Test your knowledge <quiz display="simple"> {The vmPFC contributes to fear extinction by inhibiting fear responses generated by the amygdala: |type="()"} + True - False {The amygdala is primarily responsible for forming and expressing conditioned fear responses, but extinction does not completely erase the original fear memory: |type="()"} + True - False {Fear conditioning occurs when a previously neutral stimulus becomes associated with a threatening event, causing it to trigger a conditioned fear response: |type="()"} + True - False {Fear extinction permanently erases the original fear memory, preventing the fear response from returning: |type="()"} - True + False {Fear generalisation occurs when an individual develops fear responses towards stimuli that resemble the original conditioned threat: |type="()"} + True - False {Expectancy violation occurs when an anticipated threatening outcome happens exactly as the individual predicted: |type="()"} - True + False {The hippocampus contributes to fear extinction by processing contextual information that influences whether fear or extinction memories are expressed: |type="()"} + True - False {Fear renewal occurs when a previously extinguished fear response returns following a change in environmental context: |type="()"} + True - False {Spontaneous recovery occurs when fear returns because an individual experiences a new threatening event after extinction: |type="()"} - True + False {The amygdala, vmPFC and hippocampus operate independently during fear extinction, without influencing one another: |type="()"} - True + False </quiz> ==Conclusion== Fear extinction is a complex learning process through which individuals learn that a previously threatening stimulus no longer predicts danger. Rather than erasing the original memory of fear, extinction involves the development of a new inhibitory safety association that reduces the expression of fear responses (Quirk & Mueller, 2007). Psychological processes, such as expectancy violation, prediction error, and contextual memory retrieval, contribute to this new learning by allowing individuals to update their expectations about potential threats. At a neural level, interactions between the amygdala, vmPFC, and hippocampus support regulation and retrieval of extinction memories. The amygdala contributes to fear expression, while the vmPFC supports fear inhibition and the hippocampus provides contextual information to guide memory retrieval (Maren et al., 2013). However, the persistence of original fear associations means that conditioned fear can return via renewal, reinstatement, or spontaneous recovery, highlighting that extinction memories do not always override previously learned threat. Ultimately, understanding these psychological and neural mechanisms signifies how fear extinction promotes individuals to regulate conditioned fear while keeping the original fear. ==See also == * [wikipedia:Central_nucleus_of_the_amygdala][[wikipedia:Central_nucleus_of_the_amygdala|Central Amygdala]] (Wikipedia) * [[wikipedia:Expectancy_violations_theory|Expectancy Violation]] (Wikipedia) * [[Motivation and emotion/Book/2022/Fear|Fear]] (Chapter 27, 2022) * [[wikipedia:Fear_conditioning|Fear conditioning]] (Wikipedia) * [wikipedia:Ventromedial_prefrontal_cortex][[wikipedia:Ventromedial_prefrontal_cortex|Ventromedial Prefrontal Cortex (vmPFC)]] (Wikipedia) == References == {{Hanging indent|1= Adolphs, R. (2013). The biology of fear. Current Biology, 23(2), R79–R93. National Library of Medicine. https://doi.org/10.1016/j.cub.2012.11.055 Bouton, M. E. (2004). Context and behavioral processes in extinction. Learning & Memory, 11(5), 485–494. https://doi.org/10.1101/lm.78804 Craske, M. (2015). Optimizing exposure therapy for anxiety disorders: An inhibitory learning and inhibitory regulation approach. Verhaltenstherapie, 25(2), 134–143. https://doi.org/10.1159/000381574 Craske, M. G., Kircanski, K., Moriel, Z., Mystkowski, J., Chowdhury, N., & Baker, A. (2008). Optimizing inhibitory learning during exposure therapy. Behaviour Research and Therapy, 46(1), 5–27. https://doi.org/10.1016/j.brat.2007.10.003 Dymond, S., Dunsmoor, J. E., Vervliet, B., Roche, B., & Hermans, D. (2015). Fear generalization in humans: Systematic review and implications for anxiety disorder research. Behavior Therapy, 46(5), 561–582. https://doi.org/10.1016/j.beth.2014.10.001 Furini, C., Myskiw, J., & Izquierdo, I. (2014). The learning of fear extinction. Neuroscience & Biobehavioral Reviews, 47, 670–683. https://doi.org/10.1016/j.neubiorev.2014.10.016 LeDoux, J. E. (2000). Emotion circuits in the brain. Annual Review of Neuroscience, 23(1), 155–184. https://www.annualreviews.org/docserver/fulltext/neuro/23/1/annurev.neuro.23.1.155.pdf?expires=1791735526&id=id&accname=au&checksum=66010AC47123ED188645AABCFA4C7BA7 Maren, S., Phan, K. L., & Liberzon, I. (2013). The contextual brain: Implications for fear conditioning, extinction and psychopathology. Nature Reviews Neuroscience, 14(6), 417–428. https://doi.org/10.1038/nrn3492 Milad, M. R., & Quirk, G. J. (2002). Neurons in medial prefrontal cortex signal memory for fear extinction. Nature, 420(6911), 70–74. https://doi.org/10.1038/nature01138 Milad, M. R., Wright, C. I., Orr, S. P., Pitman, R. K., Quirk, G. J., & Rauch, S. L. (2007). Recall of fear extinction in humans activates the ventromedial prefrontal cortex and hippocampus in concert. Biological Psychiatry, 62(5), 446–454. https://doi.org/10.1016/j.biopsych.2006.10.011 Myers, K. M. (2006). Different mechanisms of fear extinction dependent on length of time since fear acquisition. Learning & Memory, 13(2), 216–223. https://doi.org/10.1101/lm.119806 Quirk, G. J., & Mueller, D. (2007). Neural mechanisms of extinction learning and retrieval. Neuropsychopharmacology, 33(1), 56–72. https://doi.org/10.1038/sj.npp.1301555 Vansteenwegen, D., Hermans, D., Vervliet, B., Francken, G., Beckers, T., Baeyens, F., & Eelen, P. (2004). Return of fear in a human differential conditioning paradigm caused by a return to the original acquistion context. Behaviour Research and Therapy, 43(3), 323–336. https://doi.org/10.1016/j.brat.2004.01.001 }} ==External links== * [https://my.clevelandclinic.org/health/body/hippocampus Hippocampus] (Cleveland Clinic) * [https://pmc.ncbi.nlm.nih.gov/articles/PMC3518876/ How Prediction Errors Shape Perception, Attention, and Motivation (National Library of Medicine)] * [https://www.sciencedirect.com/science/article/abs/pii/S0005796704002347 Reinstatement of Fear Responses in Human Aversive Conditioning] (ScienceDirect) * [https://www.alleydog.com/glossary/definition.php?term=Renewal+Effect Renewal Effect] - (Alleydog) * [https://www.sciencedirect.com/topics/computer-science/spontaneous-recovery Spontaneous Recovery] (ScienceDirect) * [https://pmc.ncbi.nlm.nih.gov/articles/PMC10604397/ The Basolateral Amygdala: The Core of a Network for Threat Conditioning, Extinction, and Second-Order Threat Conditioning] (National Library of Medicine) * [https://www.sciencedirect.com/science/article/abs/pii/S0149763414002681 The learning of fear extinction] (Science Direct) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Fear]] [[Category:Motivation and emotion/Book/Learning]] qlv6ku550q7d3bk2uu6nudd6wejta61 2844040 2844038 2026-11-02T22:14:32Z Jtneill 10242 2844040 wikitext text/x-wiki {{title|Fear extinction:<br>What psychological and neural processes underlie the extinction of fear responses?}} __TOC__ ==Overview == {{RoundBoxTop|theme=3}} ;Scenario [[File:Caliber 3 protection dog.jpg|thumb|150px|'''Figure 1'''. A dog attacking a person, reinforcing fear conditioning which can help develop a fear response whenever the affected person sees a dog {{g}}]] Imagine you're bitten by a dog (see Figure 1). You might develop a fear response whenever you see a dog because the brain has learned: Dog → Danger → Fear Subsequently, if you consistently have positive encounters with dogs without any negative consequences, you will gradually learn: Dog → No Danger → Safety What are the psychological and neurological mechanisms involved? {{RoundBoxBottom}} '''Fear extinction''' is a decline in conditioned fear responses (CRs) following nonreinforced exposure to a feared conditioned stimulus (CS) (Myers, 2006). Extinction is not simply the erasure of a fear memory, it involves learning a new association that inhibits the original fear response. If an individual experiences positive outcomes while facing the fearful stimulus without receiving any maladaptive consequences you are gradually learning a process called extinction learning {{g}}. Cognitively, extinction occurs when the expected threatening outcome does not occur. Repeated safe experiences weaken the expression of fear responses and strengthen the safety association (Maren et al., 2013). However, the original fear can remain. This is reiterated by renewal, where fear returns in different circumstances, reinstatement, where fear reemerges after experiencing the threat again, and spontaneous recovery, where fear returns after a period of time (Quirk & Mueller, 2007). Several brain regions interact during fear extinction. The '''amygdala''' is essential in the attainment and expression of fear. The '''ventromedial prefrontal cortex''' (vmPFC) contributes to regulation of fear responses and the retrieval of extinction memories (Milad & Quirk, 2002). The '''hippocampus''' supplies contextual information, thus, helping decide if a previous threat should be perceived as dangerous or safe (Maren et al., 2013). Therefore, fear extinction involves new inhibitory learning rather than the erasure of the original fear memory. The interaction between hippocampus, vmPFC, and amygdala provides individuals {{missing}} the ability to learn that prior threatening stimuli can be safe, resulting in a reduction of perceived fear responses when danger is not displayed. {{RoundBoxTop|theme=1}} '''Focus questions''' * How does fear extinction change the way individuals respond to previously threatening stimuli? * What psychological processes contribute to the development of new safety associations during fear extinction? * How do the amygdala, vmPFC, and hippocampus interact to regulate fear extinction? * Why might fear responses return after successful extinction, and what does this reveal about how fear memories are stored and regulated? {{RoundBoxBottom}} == Understanding fear extinction == [[Motivation and emotion/Book/2022/Fear|Fear]] is an intervening response that assists individuals in detecting and responding to context-dependent stimuli (Adolphs, 2013). Through the use of [[wikipedia:Fear_conditioning|fear conditioning]], neutral stimuli can have a maladaptive correlation with perceived danger, {{g}} this results in a learned fear response. It is important to note that when the expected harm has failed to occur repeatedly, individuals can determine that the stimulus is not dangerous, {{g}} this is a process identified as fear extinction (Furini et al., 2014). However, extinction is not regarded as the deletion of the original fear; it is simply the formation of new safe inhibitory associations that aids in the reduction of fear. Extinction learning is influenced by contextual information, this helps identify if a past threatening stimuli should be perceived as dangerous or safe (Bouton, 2004; Quirk & Mueller, 2007; Maren et al., 2013). Having a greater understanding of these processes grants a foundation for analysing psychological and neurological mechanisms that allows fear responses to be reduced as well as explaining why fear has the potential of returning after successful extinction. === Formation of fear and fear conditioning === *'''Fear conditioning''' occurs during previous neutral stimuli becoming associated with a threatening event, consequently causing the stimulus to later trigger a fear response. This learned connection is essential for understanding how fear can be ultimately reduced through extinction (Quirk & Mueller, 2007). *When a fear has been conditioned, individuals can acquire fearful responses via maladaptive stimuli that may mirror the original danger. We can use the dog example again where an individual bitten by a dog can potentially gain a fear of other dogs, even when there is no clear danger. This is known as '''Generalisation of fear''' (Dymond et al., 2015). *'''Individual differences in fear learning''' proposes that prior experience, perceived vulnerability, and learning history can influence the intensity of the developed associated fear. These distinctions can affect how individuals respond to harmful stimuli and determine how easily they learn that the stimuli are safe. {{f}} === Fear extinction as new learning === *'''Fear Extinction''' is a result of previously threatening stimuli being repeatedly experienced without the expected threat. Instead of erasing the original fear memory, extinction allows learning new inhibitory safety associations, resulting in the individual to distinguish that the stimuli is no longer a predictive danger (Quirk & Mueller, 2007). *'''Inhibitory Associations''' occur during extinction when people learn that prior dangerous stimuli can no longer accurately predict an aversive outcome. This allows new learning to support the inhibiting of expression of the original fear response without fully erasing the original association (Bouton, 2004). *'''Repeated exposure and expectation''': Consistently facing fearful stimuli without experiencing the predicted danger can help individuals update their expectations. This process is identified as expectancy violation which is significant in inhibitory learning approaches to reducing fear (M. G. Craske et al., 2008). === The role of context and memory in extinction === *Extinction learning is influenced by the context in which it occurs. The brain uses situational information to determine if a prior threatening stimuli should be perceived as unsafe or secure. This assists in clarifying why fear can return when the stimulus has appeared during a variety of scenarios (Bouton, 2004). *Extinction memories are potentially easier to retrieve in the environment where the extinction occurred. When individuals face their previously frightening stimuli in a new environment, the original fear response may reemerge (Bouton, 2004). *Due to fear extinction not necessarily erasing the original fearful memory, fear can return following a delay in time or exposure to a threatening event. This signifies that extinction incorporates learning to regulate fear instead of simply forgetting it (Bouton, 2004). == Psychological processes underlying fear extinction == Psychological processes underlying fear extinction proposes how individuals acquire the ability to reduce fear responses when prior maladaptive stimuli lose the capability to forecast threats. Instead of forgetting fearful experiences, '''inhibitory learning''' provides new adaptive associations to rival the original fear memory (M. G. Craske et al., 2008). An integral psychological process is expectancy violation, occurring when expected danger fails to develop, thus, encouraging individuals to revise their assumptions regarding danger. It is important to note that extinction learning is influenced via situational factors, therefore, expression of fear can differ regarding the context that the stimuli is encountered in. These mechanisms demonstrate how fear responses have the ability to reduce while keeping the original fearful memory. Thus, having a greater comprehensive insight in inhibitory learning, prediction error, and contextual memory retrieval reveals how fear extinction occurs and why relapse occurs after extinction is achieved. === Inhibitory learning and the formation of safety associations === *Fear extinction is acknowledged {{f}} as a process of new inhibitory learning, instead of the complete deletion of the original fear memory. As previously stated, when a past dangerous stimuli is consistently presented without the expected maladaptive outcome, the individual begins to associate that the stimuli is safe. Let's use the dog scenario again; for example, someone who is afraid of dogs after being bitten may slowly learn through positive encounters that dogs do not necessarily anticipate danger. This new association inhibits the expression of the original fear response, allowing for adaptive behaviour regardless of the persistence of the original fear memory (Quirk & Mueller, 2007; Furini et al., 2014). *Repeated encounters with previous harmful stimuli without negative consequences can potentially reinforce safety learning. Over a period of time, individuals can become more confident that the stimulus does not always predict danger, thus reducing fear responses in common scenarios (M. G. Craske et al., 2008). *Inhibitory learning supports individuals to develop more flexible versatile responses by learning that stimuli previously associated with danger can occur without a maladaptive response. Although, this safety association may rival rather than permanently replace the original fear memory (Bouton, 2004). === Prediction error and expectancy violation === * [[wikipedia:Expectancy_violations_theory|Expectancy violation]] is an integral psychological mechanism of extinction. This occurs when the outcome an individual expects differs from what actually happens. During the occurrence of extinction, an individual may predict a maladaptive outcome but consistently experience its absence, {{g}} this is known as [https://pmc.ncbi.nlm.nih.gov/articles/PMC3518876/ prediction error]. This mismatch causes prediction error, notifying that previous expectations about threats need to be updated. Therefore, greater expectancy violation can increase extinction learning by motivating the individual to reconsider their beliefs between the relationship of the stimulus and threat (M. Craske, 2015). * '''Prediction error''' utilises information that an individual's existing expectations has the potential to contain inaccuracies. When the predicted threat consistently fails to occur, this mismatch can encourage the individual to revise their understanding of the stimulus and develop a greater association between the stimulus and safety (M. Craske, 2015) * '''Exposure experiences''' can provide more information when outcomes may differ from the expected outcome. For example, an individual expecting a dog to attack may learn that their prediction was wrong after a successful encounter, thus, having a positive correlation in the reduction of dangerous expectations during future encounters (M. G. Craske et al., 2008). === Context, memory retrieval and the return of fear === * Context and memory retrieval is another process that influences extinction learning significantly {{f}}. The learned safety association during extinction does not automatically replace the original fear association; rather, the brain must retrieve the appropriate memory depending on the context of the situation. Ultimately, fear has the potential to return through processes such as [https://www.alleydog.com/glossary/definition.php?term=Renewal+Effect renewal], when the stimulus is engaged in different contexts, or [https://www.sciencedirect.com/topics/computer-science/spontaneous-recovery spontaneous recovery], when adequate time has passed. This reinforces that successful extinction depends on both the safety of the stimulus and the ability to retrieve that safety learning when it is required (Maren et al., 2013). *'''Fear renewal''' is prominent when an extinguished fear response reemerges when an individual faces the conditioned stimuli in a different context rather than the extinction environment. This highlights that extinction memories might be context-dependent instead of being universally retrieved (Bouton, 2004). *'''Spontaneous recovery''' highlights the reemergences of prior reduced fear response following a period of time. This cognitive process implies that the original fear memory can remain after extinction, thus, highlighting the significance of maintaining and retrieving safety learning over time (Bouton, 2004). == Neural mechanisms of fear extinction == [[File:PTSD stress brain.gif|thumb|Figure 2. Regions of the limbic system and prefrontal cortex - key components for fear and memory processing.|171x171px]] Neural mechanisms of fear extinction highlights the neurological interactions amongst regions of the brain which are responsible for cognitive processes like; fear learning, emotional responses, memory retrieval and contextual processing. Despite extinction learning reducing maladaptive responses {{g}} it does not always delete the original fearful stimuli, rather, neural activity alters communication between brain regions supporting the regulation and expression of fear and the retrieval of freshly acquired extinction memories (Quirk & Mueller, 2007). There are three essential neurological structures utilised during fear extinction (see in figure 2 and 3); the amygdala, the ventromedial prefrontal cortex (vmPFC), and the hippocampus. Exploring the functions of these regions will provide a greater understanding regarding how these neural mechanisms are responsible for the acquisition and expression of fear and why previous fearful responses can reemerge. === The amygdala: acquisition and expression of fear === * The amygdala plays an integral role in the acquisition and expression of conditioned fear. During fear conditioning, the [https://pmc.ncbi.nlm.nih.gov/articles/PMC10604397/ basolateral amygdala] helps establish associations between previous neutral stimulus and an aversive outcome, additionally, the [[wikipedia:Central_nucleus_of_the_amygdala|central amygdala]] contributes to the expression of physiological and behavioural fear responses {{f}}. When fear extinction is occurring, the amygdala remains involved due to the original fear association staying. However, extinction-related neural changes allow for the reduction of fear expression when the stimulus is no longer linked to danger (Quirk & Mueller, 2007). *The amygdala contains a plethora of interconnected nuclei inside the medial temporal lobe which is associated in processing emotional stimuli. The basolateral and central regions aid in identifying threat association and planning defensive approaches (LeDoux, 2000). *Fear conditioning contains alterations in synaptic communication within the amygdala, specifically via glutamate-mediated plasticity {{f}}. When extinction occurs, GABAergic circuits assist in regulating fear-related neural activity, thus, reducing conditioned fear responses (Myers, 2006). === The vmPFC: inhibition and retrieval of extinction memories === * The [[wikipedia:Ventromedial_prefrontal_cortex|ventromedial prefrontal cortex (vmPFC)]] is vital for regulating fear following extinction learning {{f}}. Instead of removing the original fear association, the vmPFC contributes to the retrieval of newly learned safety associations and can employ inhibitory influence over amygdala-based fear responses. Literature by Quirk and Mueller (2007) identified {{how}} that activity within the medial prefrontal cortex has a positive correlation with successful extinction memory retrieval, signifying the importance in suppressing prior conditioned fear. * Neuroimaging conducted on individuals has demonstrated a positive correlation in reinforced vmPFC activity during extinction-memory retrieval with reduced conditioned fear responses {{f}}. Therefore, highlighting a positive correlation with vmPFC's involvement in recalling past learned safety information (Milad et al., 2007) {{g}}. * The '''vmPFC''' contributes to the regulation of fear via interactions with the amygdala circuits. {{what}} Evidence by Milad and Quirk (2002) implies that medial prefrontal activity can coordinate extinction-memory expression, thus, helping suppress defensive responses when previously harmful stimuli are no longer predicted to be perceived as dangerous. === The hippocampus: contextual regulation of fear extinction === [[File:Gehirn_Frontalschnitt_hippocampus.png|alt=Figure 3. The Hippocampus amongst many other regions in the brain.|thumb|218x218px|Figure 3. The Hippocampus amongst many other regions in the brain.]] * The [https://my.clevelandclinic.org/health/body/hippocampus Hippocampus] delivers significant contextual information crucial for identifying if a previously threatening stimulus should be interpreted as dangerous or safe. It has a positive relationship with the vmPFC due to its classification in which extinction learning occurred and assists in guiding retrieval of the appropriate memory. This contextual processing further explains why fear can return when a previously feared stimulus is encountered outside the extinction context, demonstrating that extinction is highly dependent on context-sensitive memory retrieval (Maren et al., 2013). * The '''hippocampus''' assists in forming representations of the environment in which fear conditioning and extinction occur. Contextual memories support distinguishing scenarios that are associated with threat from those associated with safety (Maren et al., 2013). *The hippocampus cooperates with prefrontal and amygdala networks to influence which fear-related memories are gathered. This interaction reinforces why individuals may demonstrate reduced fear in familiar environments but renewed fear in unfamiliar environments (Maren et al., 2013). '''Table 1. Brain regions neural mechanisms and their relation to fear extinction''' {{f}} {| class="wikitable" |+ !Brain Region / Mechanism !Role In Fear Extinction !Neural Mechanism |- !Amygdala |Involved in the acquisition and expression of conditioned fear, while also participating in extinction learning. |During extinction, changes within the amygdala support the learning of new extinction associations. Extinction does not simply remove the original fear association. |- !vmPFC |Supports the inhibition and retrieval of extinction memories. |The vmPFC integrates information about the stimulus and extinction learning and can exert regulatory influences over amygdala activity. |- !Hippocampus |Provides contextual information that helps determine whether a previously threatening stimulus is currently safe. |The hippocampus communicates contextual information to prefrontal and amygdala circuits, helping distinguish the extinction context from contexts associated with threat. |- !Amygdala-vmPFC Interaction |Allows the brain to regulate the expression of previously learned fear. |The vmPFC can influence amygdala circuits involved in fear expression, supporting the retrieval of the safety association. |- !Hippocampus-vmPFC Interaction |Allows extinction memories to be retrieved appropriately according to context. |Contextual information from the hippocampus helps the vmPFC determine whether extinction or fear memories should guide behaviour. |} == The return of fear following extinction == * '''Renewal: The influence of Context on Fear''' *Fear can return when a previously extinguished stimulus is encountered in a context that differs from where extinction learning occurred. This phenomenon is known as '''renewal''', it demonstrates that extinction does not erase the original fear memory {{f}}. Instead, the brain retains both the original fear association and the newer safety association, with contextual information influencing which memory is expressed. The hippocampus is important in providing contextual information that helps determine whether the stimulus should be interpreted as threatening or safe (Bouton,2004; Maren et al., 2013). === Reinstatement: the return of fear following exposure to threat === * '''[https://www.sciencedirect.com/science/article/abs/pii/S0005796704002347 Reinstatement]''' occurs when a fear response returns after an individual experiences the aversive event again following extinction. For example, if someone successfully extinguishes their fear of dogs but is subsequently bitten by a dog, their fear of dogs may return. This implies that extinction learning remains vulnerable to subsequent threatening experiences, which can reactivate or strengthen the original fear association (Quirk & Mueller, 2007; Maren et al., 2013) === Spontaneous recovery and persistance of the original fear memory === *'''Spontaneous recovery''' refers to the return of a previously extinguished fear response following the passage of time. This phenomenon provides further evidence that extinction does not permanently eliminate the original fear memory. Rather, extinction creates a competing inhibitory safety memory that can weaken or suppress fear under appropriate conditions. As time passes, the original association may regain influence, demonstrating the enduring nature of fear memories and the importance of continued extinction learning (Bouton,2004; Quirk & Mueller, 2008). {{RoundBoxTop|theme=9}} ;Case study: Return of fear through context change Vansteenwegen et al. (2004) studied human participants using the fear-conditioning paradigm. Participants first learned to associate a neutral visual stimulus with an aversive loud noise. During extinction, the stimulus was repeatedly presented without the noise, causing their fear responses to decrease. However, extinction took place in a different context from where the original fear was learned. When participants were ultimately returned to the original fear-conditioning context, their conditioned fear response returned. This was identified through an increase in electrodermal responses and increased expectations that the negative outcome would occur. Participants who experienced acquisition extinction and testing in the same context did not show the same recovery of fear. This demonstrates evidence of renewal, showing that extinction does not erase the original fear memory. Instead, extinction appears to create a new safety association that is strongly influenced by context. When the individual returns to a context associated with the original threat, the original fear memory can regain control over behaviour. This finding also highlights the importance of the hippocampus and contextual processing in determining whether fear or safety memories are retrieved. {{RoundBoxBottom}} ;Test your knowledge <quiz display="simple"> {The vmPFC contributes to fear extinction by inhibiting fear responses generated by the amygdala: |type="()"} + True - False {The amygdala is primarily responsible for forming and expressing conditioned fear responses, but extinction does not completely erase the original fear memory: |type="()"} + True - False {Fear conditioning occurs when a previously neutral stimulus becomes associated with a threatening event, causing it to trigger a conditioned fear response: |type="()"} + True - False {Fear extinction permanently erases the original fear memory, preventing the fear response from returning: |type="()"} - True + False {Fear generalisation occurs when an individual develops fear responses towards stimuli that resemble the original conditioned threat: |type="()"} + True - False {Expectancy violation occurs when an anticipated threatening outcome happens exactly as the individual predicted: |type="()"} - True + False {The hippocampus contributes to fear extinction by processing contextual information that influences whether fear or extinction memories are expressed: |type="()"} + True - False {Fear renewal occurs when a previously extinguished fear response returns following a change in environmental context: |type="()"} + True - False {Spontaneous recovery occurs when fear returns because an individual experiences a new threatening event after extinction: |type="()"} - True + False {The amygdala, vmPFC and hippocampus operate independently during fear extinction, without influencing one another: |type="()"} - True + False </quiz> ==Conclusion== Fear extinction is a complex learning process through which individuals learn that a previously threatening stimulus no longer predicts danger. Rather than erasing the original memory of fear, extinction involves the development of a new inhibitory safety association that reduces the expression of fear responses (Quirk & Mueller, 2007). Psychological processes, such as expectancy violation, prediction error, and contextual memory retrieval, contribute to this new learning by allowing individuals to update their expectations about potential threats. At a neural level, interactions between the amygdala, vmPFC, and hippocampus support regulation and retrieval of extinction memories. The amygdala contributes to fear expression, while the vmPFC supports fear inhibition and the hippocampus provides contextual information to guide memory retrieval (Maren et al., 2013). However, the persistence of original fear associations means that conditioned fear can return via renewal, reinstatement, or spontaneous recovery, highlighting that extinction memories do not always override previously learned threat. Ultimately, understanding these psychological and neural mechanisms signifies how fear extinction promotes individuals to regulate conditioned fear while keeping the original fear. ==See also == * [wikipedia:Central_nucleus_of_the_amygdala][[wikipedia:Central_nucleus_of_the_amygdala|Central Amygdala]] (Wikipedia) * [[wikipedia:Expectancy_violations_theory|Expectancy Violation]] (Wikipedia) * [[Motivation and emotion/Book/2022/Fear|Fear]] (Chapter 27, 2022) * [[wikipedia:Fear_conditioning|Fear conditioning]] (Wikipedia) * [wikipedia:Ventromedial_prefrontal_cortex][[wikipedia:Ventromedial_prefrontal_cortex|Ventromedial Prefrontal Cortex (vmPFC)]] (Wikipedia) == References == {{Hanging indent|1= Adolphs, R. (2013). The biology of fear. Current Biology, 23(2), R79–R93. National Library of Medicine. https://doi.org/10.1016/j.cub.2012.11.055 Bouton, M. E. (2004). Context and behavioral processes in extinction. Learning & Memory, 11(5), 485–494. https://doi.org/10.1101/lm.78804 Craske, M. (2015). Optimizing exposure therapy for anxiety disorders: An inhibitory learning and inhibitory regulation approach. Verhaltenstherapie, 25(2), 134–143. https://doi.org/10.1159/000381574 Craske, M. G., Kircanski, K., Moriel, Z., Mystkowski, J., Chowdhury, N., & Baker, A. (2008). Optimizing inhibitory learning during exposure therapy. Behaviour Research and Therapy, 46(1), 5–27. https://doi.org/10.1016/j.brat.2007.10.003 Dymond, S., Dunsmoor, J. E., Vervliet, B., Roche, B., & Hermans, D. (2015). Fear generalization in humans: Systematic review and implications for anxiety disorder research. Behavior Therapy, 46(5), 561–582. https://doi.org/10.1016/j.beth.2014.10.001 Furini, C., Myskiw, J., & Izquierdo, I. (2014). The learning of fear extinction. Neuroscience & Biobehavioral Reviews, 47, 670–683. https://doi.org/10.1016/j.neubiorev.2014.10.016 LeDoux, J. E. (2000). Emotion circuits in the brain. Annual Review of Neuroscience, 23(1), 155–184. https://www.annualreviews.org/docserver/fulltext/neuro/23/1/annurev.neuro.23.1.155.pdf?expires=1791735526&id=id&accname=au&checksum=66010AC47123ED188645AABCFA4C7BA7 Maren, S., Phan, K. L., & Liberzon, I. (2013). The contextual brain: Implications for fear conditioning, extinction and psychopathology. Nature Reviews Neuroscience, 14(6), 417–428. https://doi.org/10.1038/nrn3492 Milad, M. R., & Quirk, G. J. (2002). Neurons in medial prefrontal cortex signal memory for fear extinction. Nature, 420(6911), 70–74. https://doi.org/10.1038/nature01138 Milad, M. R., Wright, C. I., Orr, S. P., Pitman, R. K., Quirk, G. J., & Rauch, S. L. (2007). Recall of fear extinction in humans activates the ventromedial prefrontal cortex and hippocampus in concert. Biological Psychiatry, 62(5), 446–454. https://doi.org/10.1016/j.biopsych.2006.10.011 Myers, K. M. (2006). Different mechanisms of fear extinction dependent on length of time since fear acquisition. Learning & Memory, 13(2), 216–223. https://doi.org/10.1101/lm.119806 Quirk, G. J., & Mueller, D. (2007). Neural mechanisms of extinction learning and retrieval. Neuropsychopharmacology, 33(1), 56–72. https://doi.org/10.1038/sj.npp.1301555 Vansteenwegen, D., Hermans, D., Vervliet, B., Francken, G., Beckers, T., Baeyens, F., & Eelen, P. (2004). Return of fear in a human differential conditioning paradigm caused by a return to the original acquistion context. Behaviour Research and Therapy, 43(3), 323–336. https://doi.org/10.1016/j.brat.2004.01.001 }} ==External links== * [https://my.clevelandclinic.org/health/body/hippocampus Hippocampus] (Cleveland Clinic) * [https://pmc.ncbi.nlm.nih.gov/articles/PMC3518876/ How Prediction Errors Shape Perception, Attention, and Motivation (National Library of Medicine)] * [https://www.sciencedirect.com/science/article/abs/pii/S0005796704002347 Reinstatement of Fear Responses in Human Aversive Conditioning] (ScienceDirect) * [https://www.alleydog.com/glossary/definition.php?term=Renewal+Effect Renewal Effect] - (Alleydog) * [https://www.sciencedirect.com/topics/computer-science/spontaneous-recovery Spontaneous Recovery] (ScienceDirect) * [https://pmc.ncbi.nlm.nih.gov/articles/PMC10604397/ The Basolateral Amygdala: The Core of a Network for Threat Conditioning, Extinction, and Second-Order Threat Conditioning] (National Library of Medicine) * [https://www.sciencedirect.com/science/article/abs/pii/S0149763414002681 The learning of fear extinction] (Science Direct) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Fear]] [[Category:Motivation and emotion/Book/Learning]] r7sshgxtga2jyh5r86djw26gwsg1xhl 0 331106 2844059 2839579 2026-11-02T23:11:11Z Jtneill 10242 Copyediting 2844059 wikitext text/x-wiki {{title|Self-disclosure and emotional intimacy:<br>How does self-disclosure foster emotional closeness in relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Stockcouple.jpg|thumb|'''Figure 1'''. Image of couple hugging {{ic|Explain what this has to do with self-disclosure}/}]] ;Picture this... Jim and Pam (see Figure 1) have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”, {{g}} this confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before. Why does this self-disclosure from Jim create this emotional closeness between him and Pam? {{RoundBoxBottom}} [[wikipedia:Self-disclosure|Self-disclosure]] is the voluntary process of sharing personal thoughts, feeling and information with another person. Self-disclosure is one of the most significant and effective ways to initiate and enhance relationship intimacy, and to maintain positive personal relationships (Forgas, 2011). Intimacy or intimate come from the Latin terms ''intimus'' meaning innermost, and ''intimare'' meaning to make the innermost known (Reis, 2018). [[wikipedia:Emotional_intimacy|Emotional intimacy]] refers to a shared sense of understanding, trust and closeness within an interpersonal relationship. This chapter discusses what both self-disclosure and emotional intimacy mean, and how self-disclosures can be encouraged and responded to, with reference to psychological theories and research. {{RoundBoxTop|theme=11}} ;Focus questions [[File:Crystal Clear app ktip.svg|left|20px|]] *What is the self-disclosure? *What is emotional intimacy and why is it important? *How can self-disclosure be fostered to improve emotional intimacy in relationships? {{RoundBoxBottom}} == Self-disclosure == Self-disclosure is defined as the process of divulging personal experiences, thoughts and feelings to others (Yu & Li, 2026). Self-disclosure has long been discussed and researched for its relationship to emotional intimacy {{f}}. According to the research, we tend to like people who disclose more, we disclose more to those we like, and when we disclose, we like those we disclosed to more afterwards (Utz, 2015) {{expand}}. === Social exchange theory (Thibaut & Kelley, 1959) === [[File:Costs vs rewards.png|thumb|247x247px|'''Figure 2.''' Social Exchange Theory, costs versus rewards {{g}}|right]] [[wikipedia:Social_exchange_theory|Social Exchange Theory]] is a psychological framework that states human behaviours and relationships work through a cost-benefit analysis (see Figure 2) {{f}}. This core concept can be applied to that of self-disclosure, in which individuals assess the potential rewards and risks of sharing personal information. In the case of self-disclosure, possible rewards or benefits of divulging personal information may be validation, emotional relief and enhanced closeness {{f}}. If we assess that these are likely outcomes of self-disclosing we are more likely to do so. In contrast, potential costs or risks of self-disclosure may be a loss of privacy, judgement or fear of rejection and in this circumstance we are much less likely to self disclose. Interpersonal relationships thrive when self-disclosure remains profitable or rewarding. {{f}} ==== Norm of reciprocity ==== Within Social Exchange Theory is the concept of the [[wikipedia:Norm_of_reciprocity|norm of reciprocity]] {{f}}. This entails the universal social expectation that people should return or repay favours, gifts and acts of kindness with similar treatment {{f}}. For example, if someone gifts you an expensive item for your birthday, you will likely feel an internal obligation to get them a similarly expensive item for their birthday. This norm enables smooth, equal and fair social exchanges (Burger et al., 2009). This concept is also applicable to self-disclosure, {{g}} research has shown that self-disclosure is reciprocal in existing relationships and in the context of strangers (Sprecher et al., 2013). Hence, reciprocal self-disclosure is the process where one person shares personal thoughts, feelings or information and the other person responds with similar personal details in return. === Social penetration theory (Altman & Taylor, 1973) === [[File:Bdspt.model.webp|'''Figure 3.''' Social Penetration Theory Model (Pecune, 2013).|alt=Figure 2. Social Penetration Theory Model (Pecune, 2013).|thumb|300x300px]] [[wikipedia:Social_penetration_theory|Social Penetration Theory]] is a social psychological model that explains how interpersonal relationships evolve from superficial or surface level interactions to deep emotional intimacy through self-disclosure {{f}}. Social Penetration Theory uses an onion model (see Figure 3) as a metaphor to describe how the social penetration process works, in which individuals "peel back" others layers of personal information through social interaction (self-disclosures) {{f}}. One aspect of Social Penetration Theory is the types of information shared, which is referred to as levels {{f}}. The first {{g}} being the Superficial level or in other words, surface level information which includes visible attributes. The second level is the Intimate or peripheral level information, which is information that people usually feel free to share in any general social situation, such as their name and where they are from. Personal or Intermediate level information is the third level, in which includes information that is not always readily shared but that is also not very private, such as values and goals. Lastly, the Core or central level information is regarded as highly private and is shared when someone feels close to the other person or believes they are very trustworthy (Manning, 2019). An example of Core level information may be one's insecurities or fears. This theory states that for interpersonal relationships to develop there must be information exchanged between those within the relationship. A significant aspect of the theory is that of [[wikipedia:Social_penetration_theory#Breadth_and_depth|breadth and depth]] {{f}}. Breadth refers to the number of topics discussed, whereas depth refers to the level of intimacy of these discussions. For example, breadth may include discussing hobbies, family, interests and depth may be discussing feelings surrounding family issues or future goals (Carpenter & Greene, 2016), depth discussions may increase intimacy levels {{why}} {{g}}. According to this {{what}} theory {{g}} self-disclosure passes through phases as an interpersonal relationship progresses, {{g}} these stages include orientation, exploratory affective exchange, affective exchange, stable exchange and in some cases depenetration (see Table 1). The first stage is orientation, in which individuals only share superficial level information. Exploratory affective exchange is the second stage, where individuals may share beyond superficial information and may increase the breadth of topics discussed. The third stage is affective exchange, which includes more casual self-disclosures, communication and information from deeper levels, this stage is often reached with close friends or romantic partners. Stable exchange is the fourth stage, in which communication is open and honest, self-disclosures are continuous, and there is breadth and depth throughout conversations. This stage is maintained with few relationships such as romantic partners, close family and friends. Finally, depenetration or dissolution is a possible stage, whereby self-disclosure is reduced as a result of conflict or stress (Carpenter & Greene, 2016). {| class="wikitable" |+Table 1. The five stages of social penetration {{fact}} and examples. !Five stages of social penetration !Examples: |- |Orientation |Small talk, adhering to social norms. |- |Exploratory affective exchange |Personal attitudes & preferences. |- |Affective exchange |Casual relaxed communication, honest opinions, deeper emotional disclosures. |- |Stable exchange |Continuous open connection, deep core values, emotional reactions are predictable, breadth and depth explored. |- |Depenetration |Optional phase, communication breadth & depth withdrawal. |} == Emotional intimacy & its importance == Emotional intimacy is a part of interpersonal relationships, defined as a shared sense of closeness, trust and mutual understanding. Emotional intimacy happens between individuals when they consistently show each other who they really are and respond to each other in ways that promote feelings of safety, security and understanding. Thus, emotional intimacy can be experienced in a variety of different kind of relationships including, romantic partners, friends and family members. {{f}} === The intimacy process model (Reis & Shaver, 1988) === Reis and Shaver (1988) argue that intimacy is a dynamic, moment to moment transactional and cyclical process. The intimacy process model shows how emotional intimacy builds through a cyclical interactive progression between two individuals. This model uses self-disclosure, responsiveness, and perceived responsiveness as core components of emotional intimacy development. [[File:Ipm r and s.png|thumb|420x420px|'''Figure 4.''' The intimacy process model (Reis & Shaver, 1988).|left]] In this model {{g}} an interaction occurs between two people A and B, or for this example we will use the names Annabel and Bridgette. The model shows a single interaction in which Annabel is self-disclosing and Bridgette is responding (see Figure 4), however in most situations both individuals would freely move between self-disclosing and responding. In this example, the intimacy process begins with Annabel. Annabel has some mixture of motives, goals or fears that results in a self disclosure, whereby Bridgette must respond. As the role of the responder Bridgette is influenced by a mixture of concerns, which can be positive or negative {{g}}. Bridgette may want to be a good friend or partner or on the other hand, Bridgette may fear commitment, feel that Annabel is dependent on them and so forth. Annabel may intend {{sp}} on one thing, where Bridgette may perceive another. As Bridgette interprets the disclosure and expressions from Annabel, a complex mixture of motives and goals is at work, {{g}} this is referred to as the interpretive filter. In this model {{g}} Reis and Shaver (1988) argue that Bridgette's response to Annabel's self-disclosure is just as important as Annabel's initial disclosure, and will determine whether the interaction will become emotionally intimate. Suitable responses will enhance feelings of connectedness between the individuals, and inappropriate responses or non-responsiveness will keep the individuals at a distance. However, what matters most is Annabel's interpretation and reaction to Bridgette's response, for example if Annabel interprets Bridgette's response as unsupportive or too intrusive, greater intimacy is unlikely to be achieved (Reis & Shaver, 1988). === The importance of emotional intimacy === Emotional intimacy is intrinsically gratifying, and involves feelings of safety, trust, understanding, validation, acceptance and overall feeling cared for {{f}}. Studies have found associations between emotional intimacy, communication and relationship satisfaction, such that partners' perception of their emotional intimacy predicts both their own relationship satisfaction as well as their partners (Yoo et al., 2014). Similarly, those with strong intimate relationships tend to have higher life satisfaction (Amati et al., 2018) and wellbeing (Greene et al., 2006). Overall, intimacy can result in reciprocal intimacy, mutual liking and relationship progress (Forgas, 2011). Emotional intimacy has been found to play a role in reducing stress responses (Phillips et al., 2009). It has also been found that loneliness and depression partly stem from a lack of emotional intimacy (Rokach & Ohiliber-Lignieres, 2015). This research highlights just how important emotional intimacy is in everyday life, therefore it is important to learn how to improve emotional intimacy. {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] *Intimacy is related to relationship and life satisfaction. *Emotional intimacy is related to higher wellbeing and reduced stress responses. *Emotional intimacy can facilitate reciprocal intimacy, mutual liking and relationship progression. {{RoundBoxBottom}} ==Fostering self-disclosure== Rewarding relationships are practically inconceivable without self-disclosure. Self-disclosure is important in initiating, maintaining and enhancing emotional intimacy within interpersonal relationships, which has been linked to relationship success and satisfaction, as well as improved mental health (Forgas, 2011). Therefore, learning to foster and encourage self-disclosure is essential for everyday life. === How to encourage & respond to self-disclosures === In order to encourage self-disclosures from others we must remember Social Exchange Theory (Thibaut & Kelley, 1959) and the cost-benefit analysis. If someone wanted to disclose something to you, think to yourself, would it be rewarding for them? Making a self-disclosure conversation rewarding can be done in a multitude of ways. Social cues such as enquiries of concern (asking if someone is okay for example) or initiating with a self-disclosure displays inviting cues to the other person to disclose (Greene et al., 2006). It has also been found that moods have an effect of self-disclosure, such that positive moods increase self-disclosure intimacy, abstractness and breadth, whereas negative moods reduce intimacy but improve reciprocity of self-disclosure (Forgas, 2011), in which is an important aspect to take into account. Additionally, adherence to the norm of reciprocity, in which you can respond with a similar self-disclosure. Similarly, the intimacy process model (Reis & Shaver, 1988) highlights the underlying cause of a self-disclosure as well as the importance of responsiveness to self-disclosures. It is helpful to remember that there is an underlying motive, goal, fear or reason for someone to self-disclose. In the case of responsiveness, High or good responsiveness is comprised of the content of the response (elaborating on or matching disclosure), the style of response (showing concern and care) as well as the timing of the response (immediate or delayed). However, it is extremely important to take into account boundaries and assess the appropriateness of the situation, such as the physical environment where the interaction is occurring (Greene et al., 2006). {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] How to encourage and respond to self-disclosure: *The cost-benefit analysis: make the interaction rewarding through listening and responsiveness. *Social cues: enquiries of concern and initiating self-disclosure. *The norm of reciprocity: use reciprocal self-disclosure. *Boundaries: assess the appropriateness of the situation. {{RoundBoxBottom}} ==Conclusion== Self-disclosure creates a sense of understanding, safety, and trust that is essential for emotionally intimate interpersonal relationships. This self-disclosure process is intrinsic and crucial to progress interpersonal relationships. Self-disclosure drives emotional intimacy by creating a loop of vulnerability, validation and mutual understanding. Emotional intimacy has effects on aspects of everyday life including on communication and relationship satisfaction, such that perception of emotional intimacy can predict both one's own relationship satisfaction as well as their partners. Strong emotionally intimate relationships tend to have higher life satisfaction and wellbeing. Additionally, intimacy can result in reciprocal intimacy, mutual liking and relationship progress. Emotional intimacy has also been found to play a role in reducing stress responses and a lack of emotional intimacy can play a role in loneliness and depression. Therefore, it is important to improve emotional intimacy through learning to foster and encourage self-disclosure. Self-disclosure can be fostered through social cues, making the interaction rewarding and validating, using the norm of reciprocity and taking responsiveness into account. {{RoundBoxTop|theme=11}} ;Take home messages [[File:Crystal Clear app ktip.svg|left|20px|]] *Self-disclosure is the process of divulging personal experiences, thoughts and feelings to others. *Emotional intimacy is a shared sense of closeness, trust and mutual understanding, that can have effects on life and relationship satisfaction, wellbeing, stress and loneliness. *Self-disclosure can be fostered to improve emotional intimacy in relationships through reciprocal self-disclosure, responsiveness, and social cues. {{RoundBoxBottom}} ==See also== * [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026) * [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025) * [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia) * [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia) ==References== {{Hanging indent|1= Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000 Amati, V., Meggiolaro, S., Rivellini, G., & Zaccarin. (2018). Social relations and life satisfaction: the role of friends. ''Genus, 74''(7). 1-18. https://doi.org/10.1186/s41118-018-0032-z Burger, J. M., Sanches, J., Imberi, J. E., & Grande L. R. (2009). The norm of reciprocity as an internalized social norm: Returning favors even when no one finds out. ''Social Influence, 4''(1). 11-17. https://doi.org/10.1080/15534510802131004 Carpenter, A., & Greene, K. (2016). Social penetration theory. In C. R. Berger & M. E. Roloff (Eds.), ''The International Encyclopedia of Interpersonal Communication'' (1st ed., pp. 1-4). John Wiley & Sons. https://doi.org/10.1002/9781118540190.wbeic0160 Derlega, V. J., Winstead, B. A., & Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, & J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008 Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129 Greene, K., Derlega, V. J., & Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti & D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' (pp. 409-427). Cambridge University Press. https://doi.org/10.1017/CBO9780511606632.023 Manning, J. (2019). Thinking about interpersonal relationships and social penetration theory: Is it the same for lesbian, gay or bisexual people? In C. J. Liberman, A. S. Rancer & T. A. Avtgis (Eds.), ''Casing Communication Theory'' (pp. 293-306). Kendall Hunt Publishing Company. Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118] Phillips, A., Gallagher, S., & Carroll, D. (2009) Social support, social intimacy, and cardiovascular reactions to acute psychological stress. ''Annals of Behavioural Medicine, 37''. 38-45. http://dx.doi.org/10.1007/s12160-008-9077-0 Reis, H. T. (2018). Intimacy as an interpersonal process. In H. T. Reis (Ed.), ''Relationships, Well-Being and Behaviour: Selected works of Harry T. Reis'' (pp. 209-244). Routledge. https://doi.org/10.4324/9780203732496 Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. Duck, D. H. Fay, J. K. Simpson, & J. C. Murray (Eds.), ''Handbook of personal relationships: Theory, research and applications'' (p. 367–389). John Wiley & Sons. Rokach, A., & Philibert-Lignieres, G. (2015). Intimacy, loneliness & infidelity. ''The Open Psychology Journal, 8''. 71-77. DOI: 10.2174/1874350101508010071 Sprecher, S., Treger, S., Wondra, J. D., Hilaire, N., & Wallpe, K. (2013). Taking turns: Reciprocal self-disclosure promotes liking in initial interactions. ''Journal of Experimental Social Psychology, 49''(5). 860-866. https://doi.org/10.1016/j.jesp.2013.03.017. Tolstedt, B. E., & Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84 Utz, S. (2015). The function of self-disclosure on social network sites: Not only intimate, but also positive and entertaining self-disclosures increase the feeling of connection. ''Computers in Human Behaviour, 45''. 1-10. https://doi.org/10.1016/j.chb.2014.11.076. Yoo, H., Bartle-Haring, S., Day. R., & Gangamma, R. (2014). Couple communication, emotional and sexual intimacy, and relationship satisfaction. ''Journal of Sex & Marital Therapy, 40''(4), 275-293. https://doi.org/10.1080/0092623X.2012.751072 Yu, Y., & Li, W. (2026). A fNIRS hyperscanning study on enhancing intimacy through emotional self-disclosure from the disclosure-responder perspective. ''BMC Psychology, 14''. https://doi.org/10.1186/s40359-026-04257-3 }} ==External links== * [https://www.youtube.com/watch?v=Z4bRXozREKU&t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com) * [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Relationships]] jsjwipkuec7g711a0ppkdduw99trdrc 0 331213 2844077 2840658 2026-11-03T02:21:04Z Jtneill 10242 Adding some additional clarification tags 2844077 wikitext text/x-wiki {{title|Exercise gamification motivation:<br>How can gamification affect exercise motivation and behaviour?}} __TOC__ ==Overview== {{RoundBoxTop|theme=4}} [[File:WORKOUT.jpg|thumb|'''Figure 1.''' Man earning points for exercising.]] '''Case study: Meet Alex''' Alex recently started using a fitness app {{ic|Add link to Wikipedia that explains what this is}} to become more physically active. The app awards points for completing workouts, gives badges for reaching milestones, displays weekly challenges, and allows users to compare their progress with friends (see Figure 1). At first, Alex exercises mainly to improve their health, but soon becomes increasingly motivated by earning enough points to reach the next level. When Alex moves up the leaderboard, they feel a sense of achievement and become more motivated to exercise again. However, after falling behind other users, Alex begins to lose interest and considers stopping. Alex's experience raises an important question: why can the same gamification features that motivate exercise for some people have different effects on motivation and behaviour for others? {{RoundBoxBottom}} Regular physical activity provides important physical and psychological benefits, yet maintaining exercise behaviour can be difficult. Digital fitness technologies increasingly use gamification, such as points, rewards, challenges, feedback and social comparison, to encourage people to become more active. Gamification may make exercise more engaging by providing users with goals and feedback, but its effects on motivation and behaviour are not necessarily straightforward. Psychological science can help explain why gamification may influence exercise motivation and whether increased motivation translates into sustained exercise behaviour. Understanding motivation theories can also help identify when gamification is likely to support engagement and when it may instead reduce motivation or discourage participation. {{RoundBoxTop|theme=3}} '''Focus questions''' *What is exercise gamification and how is it used to influence motivation? *What psychological mechanisms explain how gamification can affect exercise motivation? *How does gamification influence actual exercise behaviour and adherence? *When might gamification support or undermine exercise motivation? *How can gamification be designed to encourage sustainable exercise behaviour? {{RoundBoxBottom}} == Understanding exercise gamification == Digital technologies have changed how people monitor and engage in physical activity. Smartphones, fitness applications, and wearable devices can track behaviours such as daily steps, workout frequency, and exercise intensity while providing immediate feedback on progress. These technologies also include features traditionally associated with games to make physical activity more engaging. This approach, known as gamification, is commonly used to encourage participation and maintain motivation for exercise. === What is exercise gamification? === Gamification refers to using game-design elements in non-game contexts. In exercise settings, this means adding game-like features to an activity that isn't necessarily a game. For example, a person may earn points for completing a workout, maintain a streak to reach a daily step goal, or compete against friends on a leaderboard. This differs from an exergame, where physical activity forms part of the gaming experience. Instead, gamification adds selected game elements to an existing activity to influence motivation and behaviour (Krath et al., 2021). === Common gamification features === Exercise gamification can take many forms, including points, badges, rewards, challenges, progress tracking, feedback and leaderboards. These features turn physical activity into measurable goals and achievements, allowing users to track performance and receive feedback on their progress. For example, Shameli et al. (2017) examined how app-based walking challenges could encourage users to increase their daily physical activity, demonstrating how a relatively simple game element can be applied to an everyday behaviour such as walking {{ic|What were the findings?}}. A systematic review of 50 gamified physical activity interventions found that goal-setting was the most frequently used game element, appearing in 60% of studies, followed by progress bars (52%), rewards (50%), points (44%) and feedback (42%). Competition, collaboration, leaderboards, challenges and badges were also identified, showing that gamification can involve considerably more than simply earning points or rewards (Xu et al., 2022). Importantly, different game elements may not be equally effective. A recent systematic review and meta-analysis found that the effects of gamified physical activity interventions among children and adolescents varied by factors including the game elements used, theoretical approach {{expand}}, and intervention duration (Wang et al., 2025). Gamification should therefore not be considered a single behaviour-change strategy with a consistent effect. Instead, different features may influence exercise motivation and behaviour through different psychological processes. Understanding these processes requires examining the psychological theories that explain why gamification can motivate exercise in the first place. == Psychological mechanisms of gamification == Gamification can influence exercise motivation through several psychological processes, not just by making physical activity more entertaining. Understanding these processes helps explain why the same game feature may motivate one person while pressuring or discouraging another. === Self-determination theory === Self-determination theory (SDT) provides a useful framework for understanding why the same exercise goal can feel motivating in one situation but pressuring in another. SDT proposes that motivation is influenced by the satisfaction of three basic psychological needs: autonomy (the feeling of having choice and control over one's actions), competence (the feeling of being capable and effective) and relatedness (the feeling of connection and belonging with others) (Ryan & Deci, 2020). Environments that support these needs are more likely to encourage higher-quality motivation, whereas environments that frustrate them may undermine motivation. Importantly, SDT distinguishes between different qualities of motivation rather than simply asking how motivated someone is. Autonomous motivation occurs when a behaviour is enjoyable or personally valued (Yaban & Gaschler, 2025). For example, someone may exercise because running is enjoyable or because improving their health matters to them. In contrast, controlled motivation involves internal or external pressure, such as exercising to avoid guilt, gain approval or obtain a reward (Claes et al., 2026). At the far end of the spectrum, amotivation describes a lack of intention or meaningful motivation to act (Hardcastle et al., 2015). These distinctions matter because two people can complete the same workout for very different reasons. Research supports the relevance of these distinctions to exercise behaviour. A systematic review of 66 studies found that autonomous forms of motivation were consistently associated with exercise participation, with identified regulation appearing particularly relevant to initial exercise adoption and intrinsic motivation showing stronger links with long-term adherence (Teixeira et al., 2012). Similarly, a meta-analysis involving 15,984 children and adolescents found moderate positive associations between autonomous motivation and physical activity, whereas controlled forms of motivation showed weak negative associations (Owen et al., 2014). More recent evaluations of SDT research suggest that theory-informed interventions can improve physical activity. However, effects are generally small to moderate and appear to operate mainly through autonomous motivation and competence need satisfaction (Ntoumanis & Moller, 2025). SDT therefore offers more than an explanation of whether gamification can make exercise motivating. It raises a more important question: what type of motivation gamification creates? A progress indicator that helps someone recognise improvement may support competence, whereas a reward that makes exercise feel obligatory could create external pressure. Understanding whether game features support or frustrate psychological needs may therefore help explain why gamification can strengthen motivation in some circumstances but undermine it in others. === How gamification can support motivation === Gamification supports exercise motivation when game elements help satisfy the psychological needs described by self-determination theory. Rather than motivation arising simply because an activity includes points or rewards, it matters more to understand how participants experience these features. Gamification can provide clear goals, immediate feedback, visible progression and opportunities for social interaction, all of which can contribute to motivational experiences (Krath et al., 2021). Competence may be particularly relevant to exercise gamification. Features such as progress bars, performance feedback, levels and achievable challenges allow users to see evidence of improvement. For example, watching a weekly step goal gradually fill provides immediate feedback on progress toward an achievable target. Gamification can also adjust task difficulty to a user's abilities, potentially keeping challenges achievable while still requiring effort (Krath et al., 2021). These features may help users feel increasingly capable rather than viewing exercise as an activity in which they are unsuccessful. Gamification may also support autonomy when users are given meaningful choices over their goals and how they pursue them. Personalised goals, optional challenges and different progression paths can provide greater control than systems that impose the same targets on every user. Similarly, relatedness may be supported through social features that allow users to cooperate, share progress or work towards common goals. Social interaction may therefore provide encouragement and a sense of connection alongside the exercise itself. However, the presence of these features does not necessarily mean that psychological needs will be satisfied. Reviews of health gamification have noted that interventions often combine multiple game elements, making it difficult to determine which specific features produce motivational effects (Johnson et al., 2016). This distinction matters because a leaderboard that builds connection and friendly competition for one person could create pressure or discouragement for another. Therefore, the motivational value of gamification may depend less on whether game elements are present and more on how those elements are designed and experienced. Figure 2 illustrates how game elements may influence exercise behaviour through their effects on psychological experiences and motivation. [[File:Gamification features.png|thumb|Gamification exercise motivation pathway|center|550px|'''Figure 2.''' A simplified pathway {{ic|where is the path?}} through which gamification features may influence exercise motivation and behaviour.]] <quiz display=simple> Which feature is most likely to support autonomous exercise motivation? |type="()"} - Users lose points whenever they miss a workout. || Punishing missed workouts may create pressure to exercise, making motivation feel more controlled rather than self-directed. + Users choose their own exercise goals and receive feedback about their progress. || Providing choice can support autonomy, while progress feedback can support competence. Both are important psychological needs within self-determination theory. - Users are publicly ranked against other users. || Public rankings emphasise comparison and external performance. For some users, this may shift attention towards external outcomes rather than autonomous reasons for exercising. </quiz> === When gamification can undermine motivation === <div style="border: 2px solid #4a7ebb; background-color: #f2f7fc; padding: 15px; margin: 20px 0; border-radius: 8px;"> '''Case study: Same app, different motivation''' Jordan and Sam use the same fitness app, which awards points, tracks exercise streaks, and ranks users on a leaderboard. Jordan enjoys seeing their progress and feels encouraged to exercise regularly. Sam becomes anxious about losing their streak and discouraged when they fall behind other users. '''Reflection question''' Using self-determination theory, how could the different responses of Jordan and Sam be explained? ''Think about how the app may affect each person's autonomy, competence, and reasons for exercising.'' </div> Gamification does not always support autonomous exercise motivation. From an SDT perspective, rewards, competition and performance targets may become controlling when users feel pressured to exercise for an external outcome rather than because the activity is enjoyable or personally meaningful (Ryan & Deci, 2020). For example, maintaining an exercise streak may initially encourage consistency, but the desire to avoid losing it could shift motivation toward pressure and obligation. Competition can also produce different experiences across users. Leaderboards may provide enjoyable social comparison for some people but may discourage users who repeatedly see themselves performing below others. Research on gamification similarly suggests that individual responses to game elements can vary, meaning that features intended to motivate do not necessarily benefit every user in the same way (Krath et al., 2021). This distinction highlights a potential limitation of exercise gamification. Increasing engagement with a gamified system is not necessarily the same as developing sustainable motivation for exercise. A person may exercise to earn points, maintain a streak or outperform others without developing a personally meaningful reason to remain active. Whether this motivation persists when the game element disappears therefore becomes an important question. {{f}} == Gamification and exercise behaviour == Increased motivation does not necessarily mean that a person will maintain exercise behaviour. Research must therefore distinguish between motivation, engagement with gamified features and actual changes in physical activity. === Effects on physical activity === Overall, research suggests that gamification can produce measurable increases in physical activity {{f}}. A meta-analysis of {{ic|how many studies?}} gamified smartphone applications found a small-to-moderate positive effect on physical activity compared with non-gamified conditions, indicating that game elements can influence behaviour rather than simply increasing interest in an application (Yang et al., 2022). However, substantial variation between studies suggests that gamification does not produce consistent effects across all interventions. Differences in game elements, intervention design, and participant characteristics may partly explain these mixed outcomes. === Short-term motivation and long-term adherence === Evidence that gamification increases physical activity in the short term does not necessarily mean these changes will be maintained. Mazeas et al. (2022) found that gamified interventions produced a small-to-moderate positive effect on physical activity immediately after the intervention (''g'' = 0.42). However, at a longer-term follow-up, at approximately 14 weeks, the effect was considerably smaller (''g'' = 0.15), although it remained statistically significant. This suggests that gamification may be more effective at encouraging initial engagement than producing substantial long-term behaviour change. Importantly, continued app use or pursuit of rewards should not automatically be interpreted as sustained exercise motivation. Long-term effectiveness depends not only on whether gamification encourages activity while game elements are present, but whether exercise behaviour persists over time. === Do different gamification features work differently? === The effectiveness of gamification may depend on which game elements are used and how they are designed. A systematic review and meta-analysis of {{ic|how many studies?}} gamified smartphone applications found small-to-moderate improvements in physical activity, but considerable variation between interventions (Yang et al., 2022). This variation may partly reflect differences in features such as rewards, challenges, feedback and social comparison, as well as how these elements are combined. Research also suggests that individual game elements can create different psychological experiences rather than producing a consistent motivational effect (Krath et al., 2021). Therefore, successful gamification may depend less on including as many game elements as possible and more on selecting features that support the intended motivational and behavioural outcomes. == Designing effective and sustainable exercise gamification == Understanding why gamification influences motivation can help inform how exercise technologies are designed. Effective gamification should apply psychological and behaviour-change principles rather than simply adding game elements in the hope of increasing engagement. === Supporting meaningful motivation === Effective gamification should support meaningful reasons for exercising rather than focusing only on external rewards. Features such as personalised goals, progress feedback, and achievable challenges may support autonomy and competence by giving users greater control and making improvement visible (Krath et al., 2021). From an SDT perspective, this may encourage more autonomous motivation than systems that pressure users towards predetermined targets. Gamification design should therefore consider the psychological experience created by game elements, not simply the number of features included. === Balancing rewards, competition and pressure === Rewards and competitive features require careful design because they may motivate users in different ways. Leaderboards and challenges can provide achievement and social comparison, but they may also create pressure or discourage users who consistently perform below others (Krath et al., 2021). Similarly, rewards may encourage participation without necessarily developing autonomous motivation. Exercise gamification should therefore avoid making points, streaks or rankings the primary reason for being active and instead use these features to complement personally meaningful exercise goals. === Designing for sustained exercise === Sustainable gamification should ultimately encourage continued exercise rather than dependence on the gamified system. Evidence that gamification effects become smaller at longer-term follow-up suggests that initial increases in activity do not always translate into lasting behaviour change (Mazeas et al., 2022). Future studies should also investigate which combinations and implementations of game elements are most effective, as existing studies show mixed results and considerable variation in their design (Xu et al., 2022). Gamification should therefore support meaningful goals, feedback, and psychological needs while helping exercise become a personally valued and sustainable behaviour. <div style="border: 2px solid #4f8a5b; background-color: #eef8f0;{{Text color default}}; padding: 15px; margin: 20px 0; border-radius: 8px;"> '''Key principles for sustainable exercise gamification''' ✓ '''Support meaningful goals''' — Connect gamification with personally valued reasons for exercising. ✓ '''Make progress visible''' — Provide useful feedback that supports feelings of competence. ✓ '''Provide meaningful choice''' — Allow personalisation rather than imposing identical goals on every user. ✓ '''Use rewards and competition carefully''' — Avoid creating excessive pressure or dependence on external rewards. ✓ '''Think beyond short-term engagement''' — Design gamification to support sustained exercise behaviour, not simply continued app use. </div> ==Conclusion== Gamification can influence exercise motivation and behaviour by using features such as goals, feedback, rewards, progress tracking and social interaction to make physical activity more engaging. However, simply adding game elements does not guarantee meaningful or lasting behaviour change. Self-determination theory helps explain this difference: gamification may strengthen motivation when it supports autonomy, competence, and relatedness, but may create pressure or controlled motivation when these psychological needs are frustrated. Research suggests that gamification can produce small-to-moderate increases in physical activity, although effects vary across interventions and tend to weaken over time. This highlights an important distinction between increasing short-term engagement and supporting long-term exercise adherence. Different users may experience the same game elements differently, so effective design should consider the psychological experience created rather than simply the number of features included. Sustainable exercise gamification should therefore support meaningful motivation through personalised goals, achievable challenges, useful progress feedback, meaningful choice and opportunities for social connection. Ultimately, gamification is most useful when game elements support continued physical activity rather than becoming the primary reason for exercising. ==See also== *[[Motivation and emotion/Book/2017/Gamification and motivation|Gamification and motivation]] (Book chapter, 2017) *[[Motivation and emotion/Book/2021/Physical activity motivation|Physical activity motivation]] (Book chapter, 2021) *[[Motivation and emotion/Book/2023/Physical activity tracking and exercise motivation|Physical activity tracking and exercise motivation]] (Book chapter, 2023) *[https://en.wikipedia.org/wiki/Self-determination_theory Self-determination theory] {{ic|Use an internal link as shown in Tutorial 2}} (Wikipedia) ==References== {{ic|Use italics for journal names and book titles etc.}} {{Hanging indent|1= Claes, S., De Wit, S., Le Clercq, B. L., & Annemans, L. (2026). What moves you? The relation between exercise motivation and well-being using a person-centered approach. BMC Public Health, 26(1), 497. https://doi.org/10.1186/s12889-025-26028-2 Hardcastle, S. J., Hancox, J., Hattar, A., Maxwell-Smith, C., Thøgersen-Ntoumani, C., & Hagger, M. S. (2015). Motivating the unmotivated: how can health behavior be changed in those unwilling to change? Frontiers in Psychology, 6, 835. https://doi.org/10.3389/fpsyg.2015.00835 Johnson, D., Deterding, S., Kuhn, K., Staneva, A., Stoyanov, S., & Hides, L. (2016). Gamification for health and wellbeing: A systematic review of the literature. Internet Interventions, 6, 89–106. https://doi.org/10.1016/j.invent.2016.10.002 Krath, J., Schürmann, L., & Von Korflesch, H. F. (2021). Revealing the theoretical basis of gamification: A systematic review and analysis of theory in research on gamification, serious games and game-based learning. Computers in Human Behavior, 125, 106963. https://doi.org/10.1016/j.chb.2021.106963 Mazeas, A., Duclos, M., Pereira, B., & Chalabaev, A. (2022). Evaluating the effectiveness of gamification on physical activity: systematic review and meta-analysis of randomized controlled trials. Journal of Medical Internet Research, 24(1), e26779. https://doi.org/10.2196/26779 Ntoumanis, N., & Moller, A. C. (2025). Self-determination theory informed research for promoting physical activity: Contributions, debates, and future directions. Psychology of Sport and Exercise, 80, 102879. https://doi.org/10.1016/j.psychsport.2025.102879 Owen, K. B., Smith, J., Lubans, D. R., Ng, J. Y., & Lonsdale, C. (2014). Self-determined motivation and physical activity in children and adolescents: A systematic review and meta-analysis. Preventive Medicine, 67, 270–279. https://doi.org/10.1016/j.ypmed.2014.07.033 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. Contemporary Educational Psychology, 61, 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Shameli, A., Althoff, T., Saberi, A., & Leskovec, J. (2017). How gamification affects physical activity. Proc Int World Wide Web Conf, 2017, 455–463. https://doi.org/10.1145/3041021.3054172 Teixeira, P. J., Carraça, E. V., Markland, D., Silva, M. N., & Ryan, R. M. (2012). Exercise, physical activity, and self-determination theory: A systematic review. International Journal of Behavioral Nutrition and Physical Activity, 9(1), 78. https://doi.org/10.1186/1479-5868-9-78 Wang, M., Xu, J., Zhou, X., Li, X., & Zheng, Y. (2025). Effectiveness of gamification interventions to improve physical activity and sedentary behavior in children and adolescents: Systematic review and meta-analysis. JMIR Serious Games, 13, e68151. https://doi.org/10.2196/68151 Xu, L., Shi, H., Shen, M., Ni, Y., Zhang, X., Pang, Y., Yu, T., Lian, X., Yu, T., Yang, X., & Li, F. (2022). The effects of MHealth-based gamification interventions on participation in physical activity: Systematic review. JMIR Mhealth and Uhealth, 10(2), e27794. https://doi.org/10.2196/27794 Yaban, E. H., & Gaschler, R. (2025). Autonomous motivation, self-efficacy, and developmental regulation processes in distance education across diverse age groups. American Journal of Distance Education, 40(3), 311–331. https://doi.org/10.1080/08923647.2025.2484047 Yang, Y., Hu, H., & Koenigstorfer, J. (2022). Effects of gamified smartphone applications on physical activity: A systematic review and meta-analysis. American Journal of Preventive Medicine, 62(4), 602–613. https://doi.org/10.1016/j.amepre.2021.10.005 }} ==External links== * [https://www.health.gov.au/topics/physical-activity/24-hour-movement-guidelines-for-all-australians?language=en 24-hour movement guidelines for all Australians] (Australian Government Department of Health, Disability and Ageing) * [https://www.who.int/news-room/fact-sheets/detail/physical-activity Physical activity] (World Health Organization) * [https://selfdeterminationtheory.org/topics/application-sportsandexercise/ Self-determination theory: sports and exercise] (Center for Self-Determination Theory) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Exercise]] [[Category:Motivation and emotion/Book/Gamification]] [[Category:Motivation and emotion/Book/Self-determination theory]] kitik4v704v5vusq7q2apqea57vq7yo 0 331239 2844080 2842481 2026-11-03T02:46:47Z Jtneill 10242 2844080 wikitext text/x-wiki {{title|Developing a growth mindset:<br>How can a growth mindset be cultivated and sustained?}} __TOC__ ==Overview== {{RoundBoxTop|theme=2}} [[File:Openminded.png|right|thumb|150px|'''Figure 1'''. Illustration of being open-minded]] ;Imagine this... Two childhood best friends find themselves in different places when they finish high school. Though {{g}} living in the same neighbourhood, attending the same schools, enjoying the same hobbies and receiving equal opportunities, one is earning a scholarship to study their dream degree, and the other is falling behind. How could two people with the same opportunities have vastly separate life outcomes? Well, one evaded challenge for fear of failing, while the other appraised these failures as opportunities to learn. The margin between one and the other was not created based on ability, but from mindset (see Figure 1). {{RoundBoxBottom}} Developing a growth mindset is not a single process; adopting the belief that your abilities and intelligence can change does not necessarily guarantee it will hold over time. In the face of failure, setbacks, negative feedback or difficult environmental challenges, growth-oriented beliefs may waver. Adopting maladaptive approaches to adversity can lead individuals to believe in their own incompetence and avoid challenges necessary for lifelong learning and success. The scenario raises an important question: "how can a growth mindset be sustained beyond its initial development?" Addressing this question requires understanding a growth mindset not as something linear once adopted, but as something {{vague}} that requires ongoing cultivation and reinforcement. To sustain a growth mindset beyond its initial development, we must explore the internal and external factors that cultivate and reinforce this framework over time. {{RoundBoxTop|theme=4}} '''Focus questions''' * How does a growth mindset differ from a fixed mindset? * How do mastery-oriented goal types sustain growth mindset attributes? * How does person-focused feedback inhibit the development of a growth mindset? * How can feedback be used to reinforce a growth mindset over time? * How do environmental factors, such as parental and educational interventions, inform the trajectory of growth mindset attributes over time? {{RoundBoxBottom}} == Mindsets == A [[wikipedia:Mindset|mindset]] is a mental framework or set of cognitive patterns that manages an individual's thinking, feeling and behaving, and secondary consequences of information processing and decision making (Reeve, 2024). Adopting a specific mindset creates a downstream influence on one's perception and beliefs of personal success, flaws, qualities, and other’s [[wikipedia:Belief|beliefs]] of your own [[wikipedia:Ability|abilities]]. Implementing a mindset gives way to creating altering motivational systems that manage differing ways of coping, and habits implemented on the basis of how external experiences are appraised. Dweck and Yeager (2019) suggest that mindsets are the core of meaning systems. Meaning, {{g}} mindsets allow us to create meaning systems that serve to organise [[wikipedia:Goal|goals]], beliefs and behaviours with ‘effort beliefs’ being a core concept that dictates how an individual approaches effort as a positive belief to grow or a negative fixed belief. === '''Growth versus fixed mindset's''' {{g}} === American psychologist, [[wikipedia:Carol_Dweck|Carol Dweck]], characterised that an individual can adopt two categories of mindsets, a growth or fixed mindset (Mueller & Dweck, 1988). A growth or fixed mindset’s {{g}} concern the framework in which an individual views their abilities, personal qualities and intelligence on a continuum between something that is fixed or transformative. An individual with a fixed mindset may perceive their abilities as inherent characteristics that define what they can achieve. However, an individual with a growth mindset is more likely to perceive their abilities as malleable and capable of being developed through [[wikipedia:Learning|learning]] and effort. {{g}} A person that {{g}} adopts a fixed mindset believes that their [[wikipedia:Skill|skills]], and [[wikipedia:Intelligence|intelligence]] is something that is concrete and unlikely to change (Mueller & Dweck, 1988). The [[wikipedia:Implicit_theories_of_intelligence|implicit theory of intelligence]] suggests a fixed mindset as an ‘entity theory’ leading an individual to understand their intelligence as stagnant and a product of genetic predisposition, as they perceive that these qualities are a physical body that is attached to an individual (Dweck & Leggett, 1988; Reeve, 2024). From this perspective, an individual may interpret their abilities as an intrinsic level of competence. But when faced with challenges or failure, individuals with this mindset are more likely to discern these experiences are evidence of incompetence, potentially limiting their enthusiasm to sustain or engage in challenging goal. Conversely, if a person adopts a growth mindset, they believe that with persistence, their skills and intelligence can strengthen and increase. Contradictory to an entity theorist, the implicit theory of intelligence suggests that a growth mindset is referred to as an ‘incremental theorist’, proposing that intelligence and personal capabilities are malleable traits with the possibility to improve with motivation and dedication (Dweck & Leggett, 1988). Rather than perceiving their current abilities as a fixed depiction of their greatest potential, incremental theorists perceive skills and intelligence as characteristics that can be enhanced whereby learning and effort is persistent. For an incremental theorist, though their realistic personal abilities may be limited, the possibility to improve and attain mastery is still probable. These opposing ways of thinking play a significant role in how they influence individuals to approach challenges, respond to failures and highlight opportunities for development. {{f}} === '''The role of neuroplasticity''' === [[File:Homosynaptic Plasticity-1.jpg|thumb|'''Figure 2'''''.'' Learning strengthens neural connectivity ]] From a biological perspective, the malleability of a human's mindset may not be limited to psychological explanation but can also be considered from the perspective of neural processes demonstrating that the human [[wikipedia:Brain|brain]] is capable of structural and functional change (Sarrasin et al., 2018). [[wikipedia:Neuroplasticity|Neuroplasticity]] refers to the brains {{g}} ability to alter, form new connections, and recognise pre-existing neural connections when responding to environmental demands, experience and [[wikipedia:Learning|learning]] (Ng, 2018). Meaning {{awkward}} that the brain is not a fixed structure, but with repeated learning and exposure to new experiences these practices can contribute to changes in skills and abilities. [[wikipedia:Synaptic_plasticity|Synaptic plasticity]], meaning the changes in strength between [[wikipedia:Neuron|neurons]], serving as the primary explanation for how [[wikipedia:Memory|memory]] and learning can become encoded and subsequent actions can be reproduced (see Figure 2) (Martin et al., 2000; Abbott & Nelson, 2000) {{g}}. When adopting a growth mindset, encouraging determination, seeking challenges, resilience when faced with failure, creates the conditions through which learning and synaptic plasticity transformation can occur. Neuroplasticity provides a plausible biological explanation for understanding how learning and repeated experiences, components accompanying a growth mindset, can contribute to changes in neural functioning (Sarrasin et al., 2018) {{g}}. However, though neuroplasticity provides a biological basis for which learning can occur, it cannot be inferred that adopting a growth mindset directly alters our brains on a neurological level (Ng, 2018). But {{g}} rather as a supporting feature, through which repeated experiences and learning provide an important contribution to pursue development and to strengthen a growth mindset long-term.   == Goals create patterns in behaviour == It has been highlighted that the [[wikipedia:Goal|goals]] that are integrated within a individual’s [[wikipedia:Cognition|cognitions]], behaviour and affect create the framework in which guide them to interpret and respond to experiences and events that occur (Dweck & Leggett, 1988). A goal is a task that is yet to be accomplished but strives to attain (Reeve, 2024). Individuals who adopt incremental theory, also pursue stronger learning goals, adopt more positive beliefs about effort and engage in fewer "helpless" behaviours (Blackwell et al., 2007). Those that have a well-defined, difficult, and congruent goal are more likely to direct their [[wikipedia:Mental_energy|energy]] and sustain [[wikipedia:Motivation|motivation]] to complete an objective than those without (Reeve, 2024). When adopting differing types of [[wikipedia:Goal_orientation|goal-orientations]], an individual is likely to respond and enact behavioural patterns accordingly within the bounds and purposes of the intended goal (Dweck & Leggett, 1988). When manipulating children's perceived ability of a task, Elliott and Dweck (1988) established that the goal an individual may strive for may directly alter the patterns of achievement-related behaviour, with mastery or performance-oriented behaviour being associated with subsequent achievement. Understanding the adopted goal an individual accepts provides significant importance in ensuring the maintenance of a growth mindset, as varying goal orientations predict the approaches individuals partake in when [[wikipedia:Learning|learning]], persistence in the face of challenge and responses to failure (Elliott & Dweck, 1988).   === '''Performance and Mastery Goals''' === Elliot and Dweck (1988) proposed that in order to engage in achievement behaviour, it must first be distinguished what goal an individuals {{g}} adopts when approaching a task. There are two types of goals an individual may adopt {{g}} performance or mastery goals. The approach to defining these goals is based on the perspective an individual takes when evaluating their competence and skills. Meaning {{g}} that those who take on varying goals intuitively adopt varying perceptions and reactions in different circumstances (Dweck & Leggett, 1988). Those that {{g}} adopt performance goals are those that {{g}} seek perform actions that are a testament to their competence rather than their incompetence. Those that {{g}} pursue performance goals are likely to interpret difficult circumstances as a measure of their proficiency and [[wikipedia:Aptitude|aptitude]]. Meaning {{g}} that when faced with challenges, performance-oriented individuals may experience apprehensive of negative judgement from others, with their actions seeking to demonstrate high ability, and competence in pursuit of positive evaluation (Dweck & Leggett, 1988). Within this framework, when failure threatens these goals, the likelihood that [[wikipedia:Learned_helplessness|learned helplessness]] response patterns emerge, characterised by negative self-affect and impaired performance (Seligman & Maier, 1967) {{g}}. In contrast to performance driven goals, those that {{g}} adopt [[wikipedia:Learning|learning]] or mastery-oriented goals perceive challenge as a means to seek [[wikipedia:Feedback|feedback]] and persist in tasks when they become difficult. Those that pursue mastery goals seek not to reach a goal due to external evaluation, but to acquire new skills to extend their mastery through effort and determination (Dweck & Leggett, 1988; Elliott & Dweck, 1988). Confronting failure, when enacting a mastery-oriented approach, individuals exhibit constructive self-feedback and positive self-affect. Consequently, the distinction between a mastery-oriented and performance-oriented goals is significant to understanding and informing the pattern of behaviour that may be exercised in goal pursuit to either hinder or encourage the sustainability of a growth mindset. Mastery-oriented goals provide the possibility to fuel personal beliefs that an individual is capable of learning and improvement, supportive components to prolonging the cultivation of a growth mindset. The motivational consequences of these goal orientations differ if seeking either avoidance or improvement (Elliott & Dweck, 1988), a discrepancy that is further explored in the 2 x 2 achievement goal framework (Elliot & McGregor, 2001). '''Table 1.''' ''Comparing implicit theory of intelligence, goal orientation and predicted behavioural patterns'' (Dweck & Leggett, 1988) {| class="wikitable" |+ ! !Implicit theory of intelligence !Goal Orientation !Behavioural patterns |- |'''Growth Mindset''' |Incremental theorist |Performance goal orientation (Objective is to be positively appraised on skill/s) Performance approach OR Performance avoidance goals (Elliot & McGregor, 2001) |Learned helplessness, avoidant to challenge |- |'''Fixed Mindset''' |Entity theorist |Learning/Mastery goal orientation (Objective is to develop and improve on a skill/s) Mastery approach OR mastery avoidant goals (Elliot & McGregor, 2001) |Seeks challenge, Mastery oriented, pursuing improvement on abilities |} Revising the achievement goal theory, Elliot (1999) proposed the idea of the Trichotomous Achievement Goal Framework in which suggests 3 goal types, mastery goals, performance-approach goals and performance-avoidance. Performance-approach goals involve striving to demonstrate competence relative or beyond others, whilst performance-avoidant goals are centred around refraining from being perceived as incompetent. Subsequently, Elliot & McGregor (2001) expanded on this with the 2 x 2 achievement goal framework, by differentiating mastery-approach goals and mastery-avoidance goals. Mastery-approach goals focus on developing competence, whereas mastery-avoidance goals is the pursuit to avoid task failure. Revised versions of subtypes of goals provide a nuanced perspective of how goal motivations are associated with cultivating and sustaining a growth mindset (see Table 1). For example, mastery goals allow individuals to engage in {{g}} with setbacks and to seek improvement supporting growth mindset beliefs (Elliot & McGregor, 2001; Dweck & Leggett, 1988). Goal types of individuals employ influence whether these growth mindset beliefs are translated into cognitive, behavioural and affect patterns in the context of learning and challenge. == Feedback and praise == [[wikipedia:Feedback|Feedback]] and [[wikipedia:Praise|praise]] for exhibiting high ability is often the frequent response in identifying the next actionable steps. However, praise for demonstrating competence may seem like the normalised response, this form of response may construe an individual's perception and motivation to succeed (Mueller & Dweck, 1998). Some individuals may perceive praising student [[wikipedia:Intelligence|intelligence]] as a source of motivation that enables students to pursue in learning as they internalise this feedback as true. However, the impact of feedback and praise is dependent on the information conveyed in the feedback along with how this is comprehended by the recipient (Mueller & Dweck, 1998; Haimovitz & Dweck, 2016; Gunderson et al., 2013). For instance, feedback that emphasises improvement may present the opportunity for individuals to perceive challenges as actionable potential to change, then feedback that criticises the self. === '''Person versus process feedback''' === Mueller and Dweck (1998) examined the way feedback is conveyed may lead to undermining an individuals {{g}} performance and further motivation {{rewrite}}. A seminal study by Kamins and Dweck (1999) explored how {{ic|how many?}} {{where}} children respond in scenarios involving success and failure in regard to receiving praise that was person versus process focused. Researchers had found that process-focused participants who were praised for efforts were more likely to sustain determination in the face of difficulty, respond constructively to criticism and engage in mastery behavioural patterns. Conversely, person-focused participants who received praise towards their fixed abilities showed increased likelihood to appraise failures as indicative of low ability and low self-esteem. In the way feedback may be conveyed lends itself useful in reinforcing determination and effective learning, supporting the possibility of growth (Mueller & Dweck, 1998). {{Robelbox|title=Person or process focused feedback?|theme=2}} Peter has recently finished his semester of at school. He received his report from his teacher at the end of the day with them commenting that he had “worked really hard this semester and stayed focused”. As Peter’s teacher praised his effort and concentration, they supported a growth mindset and person-focused feedback However, when he got home, he handed his academic report to his parents who said “good job! You must be so intelligent” after seeing he received straight A’s {{g}}. His parents unbeknown to person-focused feedback, supported a fixed mindset by praising his fixed ability (Mueller & Dweck, 1998). {{RoundBoxBottom}} === '''Interpreting and integrating criticism''' === [[File:Feedback-loop-general.svg|thumb|'''Figure 3.''' Feedback creates a cycle in which helps us to keep growing]] However, Haimotz and Dweck, (2016) outlined that the effect of feedback is still important but might be significantly more influential depending on how an individual interprets failure feedback. Importantly, London and Smither (2002) suggests that in regard to the maintenance of a growth mindset, feedback should not be perceived as a single event, but as a part of an ongoing reinforcement cycle (see Figure 3). The concept of positive feedback orientation refers to an individual's overall willingness to receive and respond to feedback. When given feedback, those that {{g}} perceive feedback as constructive information are more intuitively integrating this into their future behaviour and performance than those that {{g}} dismiss it (London & Smither, 2002). In experiencing ongoing feedback forming positive feedback cycles, individuals are more inclined to engage in mastery-oriented goals, adapt and apply new behaviours, accomplish tasks develop growth-oriented beliefs (Mueller & Dweck, 1988; London & Smither, 2002). However, this may only occur when external factors (e.g., [[wikipedia:Organizational_culture|workplace feedback culture]]) may provide feedback that is process-focused and clear task outcomes are establish {{g}}, in order for individuals to feel more receptive to feedback (Kamins & Dweck, 1999; London & Smither, 2002). == Impact from parents and teachers == In cultivating and sustaining a growth mindset, [[wikipedia:Parent|parents]] and [[wikipedia:Teacher|teachers]] have the important influence of creating a culture of feedback that guides the way individuals evaluate their abilities and sustain the motivation to develop (London & Smither, 2002). Haimovitz and Dweck (2016) explored how the visible mindset of a parent can be transmitted intergenerationally. Specifically, these findings demonstrate that for parents who modelled failure as a debilitating outcome, were more likely to endorse performance-oriented mindsets by acts of focusing on their children's lack of ability. Subsequently, these behaviours primed children's intelligence beliefs, by children interpreting parental failure mindsets (Haimovitz & Dweck, 2016). Furthermore, an observational study by Gunderson et al. (2018) showed that the effects of process-focused parent-child praise can have long-term implications in sustaining an incremental framework. Results indicated that process-focused praise in [[wikipedia:Toddler|toddler]] years (i.e., praise for effort and strategies) were indicative of informing growth mindsets in later childhood academic results (Gunderson et al., 2018). These findings may extend the concept of a growth mindset as not an immediate motivational response but remains relevant in altering the belief of intelligence and [[wikipedia:Academic_achievement|academic achievement]] (Macnamara & Burgoyne, 2022). [[File:05092012 - Oyster class visit Teacher Appreciation 237.jpg|thumb|'''Figure 4.''' Teachers can encourage students to engage in growth-oriented beliefs]] In addition to this, Yeager et al. (2022) suggested that the mindset-plus-supportive-context hypothesis, a theory that illustrates a student’s growth mindset can be sustained if environmental factors, such as teaching environments, allow those behaviours to flourish. Teachers have the role of conveying to student’s {{g}} corrective growth mindset attributes through modelling how mistakes can be learning opportunities and not signs of incompetence with evidence shown in reports and assignment evaluations to demonstrate continued improvement (see Figure 4) (Yeager et al., 2022). Blackwell et al. (2007) supported this through a growth mindset intervention study, investigating transitioning middle school students and the effects that an [[wikipedia:Education|education]] exercise teaching students about the malleability of intelligence and strategies for learning may demonstrate increase academic achievement in the [[wikipedia:Classroom|classroom]]. Results had shown that those students who had been taught the malleability of a mindset not only demonstrated an improvement in motivation and academic performance, but this improvement showed no decline post intervention and during challenge (Blackwell et al., 2007). Findings outline the importance that growth mindset interventions may have in the classroom. Findings suggest that teaching interventions may be particularly valuable in the longevity of a growth mindset by equipping students with practical application of growth mindset attributes derived from research and messaging.   Despite these promising findings, Macnamara and Burgoyne (2022) conducted a meta-analysis {{ic|of how many studies?}} interested in whether growth mindset interventions in educational setting legitimately increase academic achievement. Findings suggested that these growth mindset interventions did provide {{ic|how much?}} improvement to academic achievement, but it was important to consider alternative explanations for this. For example, the nature of growth mindset interventions may provide encouragement to “work harder” rather than integrating these changes into a student's mindset long-term (Macnamara & Burgoyne, 2022). Although teachers may contribute to influencing growth mindset beliefs by establishing supportive learning environments and interventions, further research is necessary in reliably understanding how these interventions translate into long-term outcomes on mindsets. ==Conclusion== Cultivating and sustaining a growth mindset extend beyond the initial adoption of the belief that intelligence and ability can be malleable (Dweck & Yeager, 2019). More though it seeks to represent the ongoing psychological processes and external environmental influences that shape how an individual approaches challenge, creates goals and reinforcing motivation to continue learning {{rewrite}}. Internally, goal orientation, interpretating and internalising feedback and failure inform the possibility of individuals engaging in growth-oriented beliefs in the face of challenge. The importance of mastery-oriented goals, and willingness to adopt learning strategies support the determination and the likelihood that ongoing improvement will be enacted. However, psychological processes do not operate independently, with environmental factors playing an important role in creating the conditions which guide individuals to engage in growth-oriented beliefs and behaviour. Environmental factors such as parental and educational influences help develop and maintain growth mindset attributes through feedback, process-focused praise with opportunities to apply this learning strategies in real-world scenarios. In bridging these insights, a growth mindset is not a perpetual state that remains stagnant once an outcome is achieved, but rather a position which requires continued cultivation, reinforcement and exercise to sustain human motivation. ==See also== * [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] (Book chapter, 2019) * [[w:Mindset|Mindsets]] (Wikipedia) * [[wikipedia:Implicit_theories_of_intelligence|Implicit theory of intelligence]] (Wikipedia) * [[wikipedia:Goal_orientation|Goal orientation]] (Wikipedia) ==References== {{Hanging indent|1= Abbott, L. F., & Nelson, S. B. (2000). Synaptic plasticity: taming the beast. Nature Neuroscience, 3(S11), 1178–1183. https://doi.org/10.1038/81453 Blackwell, L. S., Trzesniewski, K. H., & Dweck, C. S. (2007). Implicit Theories of Intelligence Predict Achievement Across an Adolescent Transition: A Longitudinal Study and an Intervention. Child Development, 78(1), 246–263. https://doi.org/10.1111/j.1467-8624.2007.00995.x Dweck, C. S., & Leggett, E. L. (1988). A social-cognitive approach to motivation and personality. Psychological Review, 95(2), 256–273. https://doi.org/10.1037/0033-295X.95.2.256 Dweck, C. S., & Yeager, D. S. (2019). Mindsets: A View From Two Eras. Perspectives on Psychological Science, 14(3), 481–494. https://doi.org/10.1177/1745691618804166 Elliot, A. J. (1999). Approach and avoidance motivation and achievement goals. Educational Psychologist, 34(3), 169–189. https://doi.org/10.1207/s15326985ep3403_3 Elliot, A. J., & McGregor, H. A. (2001). A 2 X 2 achievement goal framework. Journal of Personality and Social Psychology, 80(3), 501–519. https://doi.org/10.1037/0022-3514.80.3.501 Elliott, E. S., & Dweck, C. S. (1988). Goals: An Approach to Motivation and Achievement. Journal of Personality and Social Psychology, 54(1), 5–12. https://doi.org/10.1037/0022-3514.54.1.5 Gunderson, E. A., Sorhagen, N. S., Gripshover, S. J., Dweck, C. S., Goldin-Meadow, S., & Levine, S. C. (2018). Parent praise to toddlers predicts fourth grade academic achievement via children’s incremental mindsets. Developmental Psychology, 54(3), 397–409. https://doi.org/10.1037/dev0000444 Haimovitz, K., & Dweck, C. S. (2016). What Predicts Children’s Fixed and Growth Intelligence Mind-Sets? Not Their Parents’ Views of Intelligence but Their Parents’ Views of Failure. Psychological Science, 27(6), 859–869. https://doi.org/10.1177/0956797616639727 London, M., & Smither, J. W. (2002). Feedback orientation, feedback culture, and the longitudinal performance management process. Human Resource Management Review, 12(1), 81–100. https://doi.org/10.1016/S1053-4822(01)00043-2 Macnamara, B. N., & Burgoyne, A. P. (2023). Do growth mindset interventions impact students’ academic achievement? A systematic review and meta-analysis with recommendations for best practices. Psychological Bulletin, 149(3–4), 133–173. https://doi.org/10.1037/bul0000352 Martin, S. J., Grimwood, P. D., & Morris, R. G. M. (2000). Synaptic Plasticity and Memory: An Evaluation of the Hypothesis. Annual Review of Neuroscience, 23(1), 649–711. https://doi.org/10.1146/annurev.neuro.23.1.649 Mesler, R. M., Corbin, C. M., & Martin, B. H. (2021). Teacher mindset is associated with development of students’ growth mindset. Journal of Applied Developmental Psychology, 76, 101299. https://doi.org/10.1016/j.appdev.2021.101299 Mueller, C. M., & Dweck, C. S. (1998). Praise for intelligence can undermine children’s motivation and performance. Journal of Personality and Social Psychology, 75(1), 33–52. https://doi.org/10.1037/0022-3514.75.1.33 Ng, B. (2018). The Neuroscience of Growth Mindset and Intrinsic Motivation. Brain Sciences, 8(2), 20. https://doi.org/10.3390/brainsci8020020 Reeve, J. (2024). Understanding Motivation and Emotion. (8th ed.). John Wiley & Sons, Incorporated. Sarrasin, J. B., Nenciovici, L., Foisy, L.-M. B., Allaire-Duquette, G., Riopel, M., & Masson, S. (2018). Effects of teaching the concept of neuroplasticity to induce a growth mindset on motivation, achievement, and brain activity: A meta-analysis. Trends in Neuroscience and Education, 12, 22–31. https://doi.org/10.1016/j.tine.2018.07.003 Seligman, M. E., & Maier, S. F. (1967). Failure to escape traumatic shock. Journal of Experimental Psychology, 74(1), 1–9. https://doi.org/10.1037/h0024514 Yeager, D. S., Carroll, J. M., Buontempo, J., Cimpian, A., Woody, S., Crosnoe, R., Muller, C., Murray, J., Mhatre, P., Kersting, N., Hulleman, C., Kudym, M., Murphy, M., Duckworth, A. L., Walton, G. M., & Dweck, C. S. (2022). Teacher Mindsets Help Explain Where a Growth-Mindset Intervention Does and Doesn’t Work. Psychological Science, 33(1), 18–32. https://doi.org/10.1177/09567976211028984 }} ==External links== * [https://open.uts.edu.au/insights/career-hacks/cultivating-a-growth-mindset/ Cultivating a Growth Mindset] (UTS) * [https://www.youtube.com/watch?v=hiiEeMN7vbQ Developing a Growth Mindset with Carol Dweck] (Youtube.com) * [https://www.youtube.com/watch?v=J-swZaKN2Ic The Power of Yet | Carol S Dweck | Tedx] (Youtube.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Mindset/Growth]] i6ph9a1s94wtyx6p6p1ll8mn290fkcn 2 331333 2844129 2833292 2026-11-03T07:49:29Z Vivekidid 3108570 /* About me */ 2844129 wikitext text/x-wiki == About me == I am a Bachelor of Science in Psychology student at [https://www.canberra.edu.au/ University of Canberra]. I am interested in understanding human behaviour, motivation, emotions, and the different psychological factors that influence everyday life. As part of my studies, I am contributing to the [[Motivation and emotion/Book|Motivation and Emotion]] Wikiversity book. == Book chapter i am working on == My Book chapter is [[Motivation and emotion/Book/2026/Creative inspiration and effort|Creative inspiration and effort: How do inspiration and effort interact during the creative process?]] My book chapter focuses on creative inspiration and effort, exploring how inspiration and effort interact during the creative process. I am using Wikiversity to develop my academic writing, research, and collaborative learning skills. Through this project, I hope to improve my understanding of psychological research while contributing useful information for other learners. == Hobbies and interests == [[File:Volleyball attack and triple block-pjt.jpg|thumb|Figure 1: Outside attacker hitting the ball ]] * Badminton * Rope skipping. * Volleyball == Social Contributions == {{ic|Use a numbered list as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2025%2FWork_motivation_and_self-determination_theory&diff=2827342&oldid=2761781 Fixed Grammatical errors] [[Talk:Motivation and emotion/Book/2026/Epistemic motivation and the need for cognitive closure|Added a comment]] [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261 Shared about what I want to learn] c4ww7fw7tqfc0rdxpi9zv4lmp8qygls 1 331345 2844078 2840659 2026-11-03T02:21:14Z Jtneill 10242 2844078 wikitext text/x-wiki == Overview == Looking good. I would suggest discussing some of the literature in this section. I say this because in the template it says to outline "psychological science". [[User:Jack4234|Jack4234]] ([[User talk:Jack4234|discuss]] • [[Special:Contributions/Jack4234|contribs]]) 08:14, 20 August 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:U3260591|Hi [[User:U3260591|U3260591]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:01, 30 August 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions <!-- Focus questions --> # Reasonably good alignment between focus questions and heading structure, but consider closer alignment <!-- Key points--> <!-- Overall --> # Solid development <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # Select the best theories about this topic # Select the best research about this topic <!-- Conclusion --> # Conclusion is underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Relevant figure(s) presented, captioned, and cited <!-- Caption --> # Figure caption(s) should include '''Figure X'''. ... <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Very good <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] |8= <!-- Resources --> <!-- See also --> # See also ## Very good ## Use internal link style for Wikipedia links (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- External links --> # External links ## Excellent ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |9= <!-- User page --> # Used effectively # Brief description about self – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # The Wikiversity talk page link goes an edit rather than read interface }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:01, 30 August 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # Excellent use of SDT in a critically balanced way, with integrated review of research, to explain the pros and cons of exercise gamification # The chapter could be enhanced by integrating more strongly with other chapters by using embedded links to related topics # Provide more detailed edit summaries to communicate with others about what was changed <!-- Overall - GenAI --> # In some places, I suspect there may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|genAI output]] <!-- Overall – Citations --> # Excellent use of academic, peer-reviewed citations to support claims <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FExercise_gamification_motivation&diff=2840658&oldid=2836631 these copyedits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Feature --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature --> # Scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent use is made of self-determination theory # This section could be strengthened by also pointing to alternative theoretical perspectives <!-- Theory – Builds on --> # Doesn't build on [[w:|Wikipedia]] articles or related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Very good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Very good use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used <!-- Theory – Examples --> # Very good use of examples to illustrate theoretical concepts # What are some examples of well-designed exercise gamification apps? |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses <!-- Research – Critical thinking --> # Very good [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies |5= <!-- Integration comments... --> # Excellent integration between the most relevant theory(ies) and the best research # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent summary and conclusion # Key principles feature box is excellent # Remind the reader about the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellen <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections <!-- Written expression – Grammar --> # Grammar and spelling are excellent <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] <!-- Written expression – Figures --> ## Figures ### Well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] <!-- Written expression – References --> ## References use very good APA style: ### Check and correct use of italicisation <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Very good use of learning features <!-- Learning features – Wikipedia embedded links --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Reasonably good use of figure(s) # No use of table(s) # Good use of feature box(es) # Basic use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Basic use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] # Basic use of the [[{{PAGENAME}}#External links|External links]] section |9= <!-- Social contribution comments... --> # ~3-4 logged, useful, mostly minor contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~3-4 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:10, 16 October 2026 (UTC) jkdc3iif6xaps9gwnqx5cbi9dvcf7fo 1 331432 2844060 2830700 2026-11-02T23:12:05Z Jtneill 10242 https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-disclosure_and_emotional_intimacy&diff=2844059&oldid=2839579 2844060 wikitext text/x-wiki == Scenario Suggestion - From U3275775 == For your overview, I think you could add a scenario about a coffee shop worker and their regular customer who comes in everyday. At first they would only make small talk but as the customer comes in each day, their talks get longer. This would demonstrate how self-disclosure develops gradually over-time, and how this leads to emotional intimacy. [[User:U3275775|U3275775]] ([[User talk:U3275775|discuss]] • [[Special:Contributions/U3275775|contribs]]) 04:56, 24 August 2026 (UTC) == Resource == Hello! I am loving the start to this topic it looks good so far and I'm interested to see how it develops. I found a resource that may be helpful, it's an article about brain hyperscanning to better understand the brain mechanisms that underlie emotional intimacy through self-disclosure. I'll paste the link and I hope it is helpful or at least points you to some extra resorces. Cheers! https://link.springer.com/article/10.1186/s40359-026-04257-3 [[User:J.M.A Watson|J.M.A Watson]] ([[User talk:J.M.A Watson|discuss]] • [[Special:Contributions/J.M.A Watson|contribs]]) 21:54, 27 August 2026 (UTC) == Extra Resource == Hey, great job so far! I noticed out topics had some similarities and thought you might benefit from this source I've had a look at :) Reis, H. T., & Shaver, P. (1988). <nowiki>''</nowiki>Intimacy as an Interpersonal Process<nowiki>''</nowiki> (pp. 367–389). Routledge. <nowiki>https://www.researchgate.net/publication/347687013_Intimacy_as_an_interpersonal_process</nowiki> [[User:U3282586|U3282586]] ([[User talk:U3282586|discuss]] • [[Special:Contributions/U3282586|contribs]]) 10:55, 28 August 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:U3283302|Hi [[User:U3283302|U3283302]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:28, 31 August 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings <!-- Alignment with focus questions --> # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement |3= <!-- Overview--> # Scenario, image, evocative description of the problem/topic, and focus questions <!-- Description --> # A promising description of the problem/topic is planned or presented <!-- Focus questions --> # Reasonably good alignment between focus questions and heading structure, but consider closer alignment |4= <!-- Key points--> <!-- Overall --> # Promising development - basic development of bullet-points with some citations for the proposed sections # Highlight the most relevant theories and synthesise the best research on the topic # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) <!-- Conclusion --> # Conclusion is well underway |5= <!-- Figure --> # Relevant figure(s) are presented and captioned <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Very good # Only cite sources you have consulted <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## use dois where available instead of other links ## include hyperlinked dois |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Used effectively <!-- Description about self --> # Brief description about self – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:28, 31 August 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic chapter; it is strong in theory, weak in research, and moderate in style/expression # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic #* quality of written expression #* integrate more strongly with other chapters by using embedded links to related topics <!-- Overall - Edit summaries --> #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall – Citations --> # Basic use of academic, peer-reviewed citations to support claims <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-disclosure_and_emotional_intimacy&diff=2844059&oldid=2839579 these copy edits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # The Figure 1 caption could be more relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Very good—key theories are well explained and applied; minor areas for improvements <!-- Theory – Builds on --> # Builds in a basic way on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Very good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. <!-- Theory – Examples --> # Very good/ use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Good summary and conclusion # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Clear take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is basic <!-- Style – Sentences --> ## The clarity and grammar of many sentences could be improved; consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. <!-- Style – Layout --> # Layout ## The chapter structure is underdeveloped ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Remove abbreviations/citations from headings <!-- Style – Grammar --> # Grammar ## The grammar for many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) <!-- Style – Figures --> ## Figures ### Reasonably well captioned <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Style – Tables --> ## Table(s) ### Well captioned <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Style – Citations --> # In many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses <!-- Style – References --> ## References use very good to excellent APA style: ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent use of learning features <!-- Learning features – Wikipedia embedded links --> # Reasonably good use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent use of figure(s) # Very good use of table(s) # Very good use of feature box(es) # Very good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent use of the [[{{PAGENAME}}#See also|See also]] section <!-- Learning features – External links --> # Insufficient use of the [[{{PAGENAME}}#External links|External links]] section ## Link to the top 3-6 external resources about this topic |9= <!-- Social contribution comments... --> # ~2 logged, useful, mostly moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~1 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # For additional social contribution credit, list uploads to Wikimedia Commons. ## This file is poorly named: [[:File:Ipm r and s.png]] (I've requested that it be renamed) }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:12, 2 November 2026 (UTC) t2b45az5ou9uxa5vvc34jdfl87xo5tv 2844064 2844060 2026-11-02T23:16:17Z Jtneill 10242 2844064 wikitext text/x-wiki == Scenario Suggestion - From U3275775 == For your overview, I think you could add a scenario about a coffee shop worker and their regular customer who comes in everyday. At first they would only make small talk but as the customer comes in each day, their talks get longer. This would demonstrate how self-disclosure develops gradually over-time, and how this leads to emotional intimacy. [[User:U3275775|U3275775]] ([[User talk:U3275775|discuss]] • [[Special:Contributions/U3275775|contribs]]) 04:56, 24 August 2026 (UTC) == Resource == Hello! I am loving the start to this topic it looks good so far and I'm interested to see how it develops. I found a resource that may be helpful, it's an article about brain hyperscanning to better understand the brain mechanisms that underlie emotional intimacy through self-disclosure. I'll paste the link and I hope it is helpful or at least points you to some extra resorces. Cheers! https://link.springer.com/article/10.1186/s40359-026-04257-3 [[User:J.M.A Watson|J.M.A Watson]] ([[User talk:J.M.A Watson|discuss]] • [[Special:Contributions/J.M.A Watson|contribs]]) 21:54, 27 August 2026 (UTC) == Extra Resource == Hey, great job so far! I noticed out topics had some similarities and thought you might benefit from this source I've had a look at :) Reis, H. T., & Shaver, P. (1988). <nowiki>''</nowiki>Intimacy as an Interpersonal Process<nowiki>''</nowiki> (pp. 367–389). Routledge. <nowiki>https://www.researchgate.net/publication/347687013_Intimacy_as_an_interpersonal_process</nowiki> [[User:U3282586|U3282586]] ([[User talk:U3282586|discuss]] • [[Special:Contributions/U3282586|contribs]]) 10:55, 28 August 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:U3283302|Hi [[User:U3283302|U3283302]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:28, 31 August 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings <!-- Alignment with focus questions --> # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement |3= <!-- Overview--> # Scenario, image, evocative description of the problem/topic, and focus questions <!-- Description --> # A promising description of the problem/topic is planned or presented <!-- Focus questions --> # Reasonably good alignment between focus questions and heading structure, but consider closer alignment |4= <!-- Key points--> <!-- Overall --> # Promising development - basic development of bullet-points with some citations for the proposed sections # Highlight the most relevant theories and synthesise the best research on the topic # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) <!-- Conclusion --> # Conclusion is well underway |5= <!-- Figure --> # Relevant figure(s) are presented and captioned <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Very good # Only cite sources you have consulted <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## use dois where available instead of other links ## include hyperlinked dois |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Used effectively <!-- Description about self --> # Brief description about self – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:28, 31 August 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic chapter; it is strong in theory, weak in research, and moderate in style/expression # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic #* quality of written expression #* integrate more strongly with other chapters by using embedded links to related topics #* more substantial engagement in social contribution <!-- Overall - Edit summaries --> #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall – Citations --> # Basic use of academic, peer-reviewed citations to support claims <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-disclosure_and_emotional_intimacy&diff=2844059&oldid=2839579 these copy edits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # The Figure 1 caption could be more relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Very good—key theories are well explained and applied; minor areas for improvements <!-- Theory – Builds on --> # Builds in a basic way on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Very good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. <!-- Theory – Examples --> # Very good/ use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Good summary and conclusion # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Clear take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is basic <!-- Style – Sentences --> ## The clarity and grammar of many sentences could be improved; consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. <!-- Style – Layout --> # Layout ## The chapter structure is underdeveloped ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Remove abbreviations/citations from headings <!-- Style – Grammar --> # Grammar ## The grammar for many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) <!-- Style – Figures --> ## Figures ### Reasonably well captioned <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Style – Tables --> ## Table(s) ### Well captioned <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Style – Citations --> # In many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses <!-- Style – References --> ## References use very good to excellent APA style: ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent use of learning features <!-- Learning features – Wikipedia embedded links --> # Reasonably good use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent use of figure(s) # Very good use of table(s) # Very good use of feature box(es) # Very good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent use of the [[{{PAGENAME}}#See also|See also]] section <!-- Learning features – External links --> # Insufficient use of the [[{{PAGENAME}}#External links|External links]] section ## Link to the top 3-6 external resources about this topic |9= <!-- Social contribution comments... --> # ~2 logged, useful, mostly moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~1 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # For additional social contribution credit, list uploads to Wikimedia Commons. ## This file is poorly named: [[:File:Ipm r and s.png]] (I've requested that it be renamed) }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:12, 2 November 2026 (UTC) mclc3rgoe9ingbp07zrc46uhduoln5o 0 331450 2843978 2831657 2026-11-02T13:01:41Z Diaz Chas 3109441 Added where learning figures should be added 2843978 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} * '''Define enjoyment in learning''': briefly define enjoyment and how it relates to learning. * '''Why is it important''': explore whether enjoyment may influence students motivation, engagement, and learning results. * '''How can psychology help''': Use psychological theories and research journal articles or book chapters on motivation, and learning to understand how and why enjoyment may affect learning. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == * What enjoyment means when it comes to [[learning]]. * How enjoyment can be a positive emotion during learning. * The difference between actually enjoying learning and just having fun. * What psychological theory can explain enjoyment in learning - (Mansfield, 2025) --- Add a quiz or a learning figure == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' --- Add a quiz or a learning figure{{Robelbox|theme=5|title=Test yourself!}} <div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} - It may encourage them to become more involved in learning activities. - It always guarantees higher marks. + It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == {{ic|Include an introductory paragraph before branching into sub-sections}} === Cognitive processes === * How enjoyment can affect [[w:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[w:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == Provide 6 APA style citations and references to key peer-reviewed source. {{Hanging indent|1= Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] bhzn5sgsv5jw10mvlyl7ghg1bqdjuhg 2843982 2843978 2026-11-02T14:25:32Z Diaz Chas 3109441 Added two additional references. ChatGPT - 3 Nov 2026. Used to help with structuring and organising the paragraph, helped brainstorm ideas on what to include, and helped polish my written draft for readability, spelling, and grammar. 2843982 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario.{{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} * '''Define enjoyment in learning''': briefly define enjoyment and how it relates to learning. * '''Why is it important''': explore whether enjoyment may influence students motivation, engagement, and learning results. * '''How can psychology help''': Use psychological theories and research journal articles or book chapters on motivation, and learning to understand how and why enjoyment may affect learning. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == * What enjoyment means when it comes to [[learning]]. * How enjoyment can be a positive emotion during learning. * The difference between actually enjoying learning and just having fun. * What psychological theory can explain enjoyment in learning - (Mansfield, 2025) --- Add a quiz or a learning figure == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' --- Add a quiz or a learning figure{{Robelbox|theme=5|title=Test yourself!}} <div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} - It may encourage them to become more involved in learning activities. - It always guarantees higher marks. + It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == {{ic|Include an introductory paragraph before branching into sub-sections}} === Cognitive processes === * How enjoyment can affect [[w:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[w:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] 85s6z6yiaqbul2hu87my1xpgbhksq7c 2843985 2843982 2026-11-02T15:11:17Z Diaz Chas 3109441 Added 2 references. ChatGPT, 3 Nov 2026 - used to simplify ideas, structure and organise the paragraph, and polish the draft for readability. 2843985 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario.{{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results.{{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == * What enjoyment means when it comes to [[learning]]. * How enjoyment can be a positive emotion during learning. * The difference between actually enjoying learning and just having fun. * What psychological theory can explain enjoyment in learning - (Mansfield, 2025) --- Add a quiz or a learning figure == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' --- Add a quiz or a learning figure{{Robelbox|theme=5|title=Test yourself!}} <div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} - It may encourage them to become more involved in learning activities. - It always guarantees higher marks. + It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == {{ic|Include an introductory paragraph before branching into sub-sections}} === Cognitive processes === * How enjoyment can affect [[w:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[w:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] ehdxrdplm2w34t9m7rpnlamnary5ckd 2843986 2843985 2026-11-02T15:22:40Z Diaz Chas 3109441 /* Enjoyment in learning */ divided into smaller sections 2843986 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario.{{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results.{{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === - What enjoyment means in learning. 3-5 sentences. 80-100 words === Enjoyment as an Emotion === - How enjoyment can be a positive emotion during learning. 3-5 sentences. 80-100 words === Enjoyment vs Fun === - the difference between enjoying learning and simply having fun. 3-5 sentences. 80-100 words ---What psychological theory can explain enjoyment in learning - (Mansfield, 2025) --- Add a quiz or a learning figure == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' --- Add a quiz or a learning figure{{Robelbox|theme=5|title=Test yourself!}} <div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} - It may encourage them to become more involved in learning activities. - It always guarantees higher marks. + It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == {{ic|Include an introductory paragraph before branching into sub-sections}} === Cognitive processes === * How enjoyment can affect [[w:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[w:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] mnjbkqsyx2i5j17k09syzgpewbu7tvp 2844085 2843986 2026-11-03T03:06:04Z Diaz Chas 3109441 Added more references, fixed an issue with formatting, added a scenario and a figure. CHatGPT, 3 Nov 2026 - used for initial brainstorming, ideas for the information discussed in each sentence, and some polishing. 2844085 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results.{{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? </div> {{-}}{{Robelbox/close}} === Enjoyment vs Fun === - the difference between enjoying learning and simply having fun. 3-5 sentences. 80-100 words ---What psychological theory can explain enjoyment in learning - (Mansfield, 2025) --- Add a quiz or a learning figure == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' --- Add a quiz or a learning figure{{Robelbox|theme=5|title=Test yourself!}} <div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} - It may encourage them to become more involved in learning activities. - It always guarantees higher marks. + It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] 2sv7d2tzqyumdkyv8p4kts67uqe1hrz 2844087 2844085 2026-11-03T03:43:08Z Diaz Chas 3109441 /* What is enjoyment in learning? */ Added a reference and quiz. ChatGPT - used to structure the sequence of the information, and helped polish the draft. 2844087 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? </div> {{-}}{{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] oy37ekzs08lsgn23v6puaey4klmevf3 2844089 2844087 2026-11-03T03:45:46Z Diaz Chas 3109441 /* Enjoyment as an Emotion */ fixed a formatting issue 2844089 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? </div> {{-}}{{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] cbdd9j6357ieanjszdfmk4suf2thayx 2844091 2844089 2026-11-03T03:46:29Z Diaz Chas 3109441 /* Enjoyment as an Emotion */ fixed a formatting issue 2844091 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}}{{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] 4pn9w6v6sk2y1v83k0fgiy2zmyg1v07 2844092 2844091 2026-11-03T03:48:12Z Diaz Chas 3109441 trying to edit format 2844092 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] k83fasjweqpay50cmgr83iqey8n0ivl 2844093 2844092 2026-11-03T03:48:58Z Diaz Chas 3109441 /* Enjoyment as an Emotion */ 2844093 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] sleqzk2gqch9xxjgjfilwzioosebzj5 2844094 2844093 2026-11-03T03:49:29Z Diaz Chas 3109441 /* What is enjoyment in learning? */ spacing 2844094 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] 9yrtq79f3hnsytf9xibfv1n4w22p4ko 2844095 2844094 2026-11-03T03:49:53Z Diaz Chas 3109441 /* Understanding Enjoyment and Learning */ spacing 2844095 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] tnnmspow1zpm6ixusm2iqq9jt5sx8s8 2844096 2844095 2026-11-03T03:50:13Z Diaz Chas 3109441 /* Enjoyment vs Fun */ spacing 2844096 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' --- Add a quiz or a learning figure == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] 73gg5aeewdifikm9jgwevozfogoj68b 2844097 2844096 2026-11-03T04:01:13Z Diaz Chas 3109441 /* Factors that influence enjoyment in learning */ Added sub sections 2844097 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == === Confidence and Ability === === Personal Interest === === Perceived value and Usefulness === === Teachers and Learning Environment === --- Add a quiz or a learning figure * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] qag2i1ehbjjj152eehonjkapcsqbvbg 2844115 2844097 2026-11-03T06:25:38Z Diaz Chas 3109441 /* Confidence and Ability */ Added a table 2844115 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Intro paragraph {| class="wikitable" |+Factors !Factors !Explanation !Example |- |Confidence and ability |How a person's enjoyment and confidence be influenced by feeling capable when completing a task | |- |Personal interest | | |- |Usefulness | | |- |Teaching methods | | |- |Learning environment | | |} === Confidence and Ability === === Personal Interest === === Perceived value and Usefulness === === Teachers and Learning Environment === --- Add a quiz or a learning figure * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] nlh99d1q86yi97dafi7ztmw6vvxhwr3 2844116 2844115 2026-11-03T06:27:49Z Diaz Chas 3109441 testing how to restrict table size 2844116 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Intro paragraph {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 25%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 25%;" | Example |- | stuff | more stuff | stuff |- | Row 2, Cell 1 | Row 2, Cell 2 | Row 2, Cell 3 |} stuff {| class="wikitable" |+Factors !Factors !Explanation !Example |- |Confidence and ability |How a person's enjoyment and confidence be influenced by feeling capable when completing a task | |- |Personal interest | | |- |Usefulness | | |- |Teaching methods | | |- |Learning environment | | |} === Confidence and Ability === === Personal Interest === === Perceived value and Usefulness === === Teachers and Learning Environment === --- Add a quiz or a learning figure * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] 4jj88fnmq8a746s2q32ois2nnl3tr8j 2844117 2844116 2026-11-03T06:28:59Z Diaz Chas 3109441 /* Factors that influence enjoyment in learning */ readjusting table 2844117 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Intro paragraph {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 20%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 30%;" | Example |- | stuff | more stuff | stuff |- | Row 2, Cell 1 | Row 2, Cell 2 | Row 2, Cell 3 |} stuff {| class="wikitable" |+Factors !Factors !Explanation !Example |- |Confidence and ability |How a person's enjoyment and confidence be influenced by feeling capable when completing a task | |- |Personal interest | | |- |Usefulness | | |- |Teaching methods | | |- |Learning environment | | |} === Confidence and Ability === === Personal Interest === === Perceived value and Usefulness === === Teachers and Learning Environment === --- Add a quiz or a learning figure * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] 46mkno8pkujd4a4flfb2t6gps8939z5 2844118 2844117 2026-11-03T06:30:11Z Diaz Chas 3109441 /* Factors that influence enjoyment in learning */ readjusted table 2844118 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Intro paragraph {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- | | more stuff | stuff |- | Row 2, Cell 1 | Row 2, Cell 2 | Row 2, Cell 3 |} stuff {| class="wikitable" |+Factors !Factors !Explanation !Example |- |Confidence and ability |How a person's enjoyment and confidence be influenced by feeling capable when completing a task | |- |Personal interest | | |- |Usefulness | | |- |Teaching methods | | |- |Learning environment | | |} === Confidence and Ability === === Personal Interest === === Perceived value and Usefulness === === Teachers and Learning Environment === --- Add a quiz or a learning figure * How can feeling confident or capable of doing something affect enjoyment. * How can being interested in what you are learning make it more enjoyable - (Hernik & Jaworska, 2018) * How can finding something useful or important affect enjoyment. * How teachers, activities, or the learning environment make learning more or less enjoyable - (Closs et al., 2021) * Find research about what can affect '''how much students enjoy learning.''' == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Closs, L., Mahat, M., & Imms, W. (2021). Learning environments’ influence on students’ learning experience in an Australian Faculty of Business and Economics. Learning Environments Research, 25(1). https://link.springer.com/article/10.1007/s10984-021-09361-2 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] rr53jni6zymwv5ld4100u8iokpna1aa 2844131 2844118 2026-11-03T08:32:36Z Diaz Chas 3109441 /* Factors that influence enjoyment in learning */ Added more references, finalised the table, fixed some formatting, and added a quiz. ChatGPT, 3 Nov 2026 - used for initial brainstorming, sentence sequencing, and topic ideas. 2844131 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Enjoyment in learning can be influenced by several different factors. Factors such as confidence, and interest are personal characteristics that may affect how much a person enjoys learning (Sabaratnam, 2025). Other factors such as the usefulness of the knowledge, the atmosphere, and the teaching methods can also influence enjoyment (Moghadasi & Keikavoosi-Arani, 2023). These factors can influence a person's motivation, participation and their overall learning experience. The table below outlines some key factors that may influence enjoyment in learning and some examples on how they can affect an individual. {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- |Confidence and ability | How enjoyment and confidence be influenced by the feeling of capability when completing a learning task. | A student feels satisfied after successfully completing an assignment. |- | Personal interest | How a person's interest in a topic increase curiosity and encourage them to learn more. | A student enjoys playing the flute as they are interested in joining the concert band. |- |Perceived usefulness |How realising the usefulness of knowledge make can make learning more meaningful. |A player enjoys learning about the different attacks in volleyball as they can apply that knowledge during the game. |- |Teaching methods |How the different methods such as activities and approches can encourage students willingness to participate. |A student enjoys the chemistry experiment after learning the ways chemicals can mix together. |- |Learning environment |How the atmosphere of the room, teachers support, and student interactions can increase enjoyment. |A student feels comfortable sharing ideas as the teacher encourages participation. |} === Research on Enjoyment in Learning === Research shows how personal characteristics and environmental factors influence student's enjoyment and their learning experiences. When a person has an interest in a subject or topic, they are more likely to learn more, pay attention, and retain the information. According to Harackiewicz et al. (2016), students who have an interest in a subject can increase their concentration, and their willingness to learn more. This shows that personal interest in a topic can lead to stronger understanding, higher participation, and more positive learning experiences. Enjoyment can also be influenced by the environment the person is in which can result in more retained information. According to Hernik and Jaworska (2018), students who attended a class that had a more enjoyable atmosphere illustrated a better recall and happiness. Additionally, research also shows that different teaching approaches and social activities can result in higher participation rates among students (Bonney, 2015). This shows that the classroom conditions, teaching approaches and social aspects can greatly contribute to a higher retention rate, and a more enjoyable learning experience. Let's look at another scenario. {{Robelbox|title=Scenario|iconwidth=55px|theme=5}}<div style="{{Robelbox/pad}}"> [[File:Hands raised (7645867).jpg|thumb|226x226px|Figure 3. Students in the classroom raised their hands while the teacher picked a student.]] Linda is learning about the chemical compounds in a science laboratory, and she loves it. Their teacher breaks down each concept into small, manageable chunks and encourages students to participate. Poppy is learning maths in a computer lab, but she is not particularly interested in maths. The teacher explains the steps it takes to solve a problem and quickly instructs the students to answer math problems using the computers. However, Poppy got bored and ended up playing games for the remainder of the session. Which student do you think had a higher recall of the topic in the next lesson? <div>{{Robelbox/close}}{{Robelbox|width=75%|theme=4|iconwidth=55px|title=Test your knowledge!}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. Which of the following is an example of how a learning environment can affect a student's retention? |type="()"} + A student retains information more easily because the classroom was quiet. - QA student enjoys learning because they are interested in the subject. - A student feels confident because they understand he topic. </div> </quiz> </div>{{Robelbox/close}} == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bonney, K. M. (2015). Case study teaching method improves student performance and perceptions of learning gains. Journal of Microbiology & Biology Education, 16(1), 21–28. https://doi.org/10.1128/jmbe.v16i1.846 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Moghadasi, J., & Keikavoosi-Arani, L. (2023). Investigating the factors influencing students’ academic enthusiasm for a shift of paradigm among education managers shaping academic pedagogy. BMC Medical Education, 23(1), 480. PubMed Central. https://doi.org/10.1186/s12909-023-04453-4 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Sabaratnam, A. (2025). The impact of students’ self-confidence on their learning interest. International Journal of Science and Research Archive, 15(2), 1632–1641. https://doi.org/10.30574/ijsra.2025.15.2.1599 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] prbjlcdoeabejzo4l0wmtyh0h941jf6 2844132 2844131 2026-11-03T08:33:09Z Diaz Chas 3109441 /* Enjoyment as an Emotion */ spacing 2844132 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Enjoyment in learning can be influenced by several different factors. Factors such as confidence, and interest are personal characteristics that may affect how much a person enjoys learning (Sabaratnam, 2025). Other factors such as the usefulness of the knowledge, the atmosphere, and the teaching methods can also influence enjoyment (Moghadasi & Keikavoosi-Arani, 2023). These factors can influence a person's motivation, participation and their overall learning experience. The table below outlines some key factors that may influence enjoyment in learning and some examples on how they can affect an individual. {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- |Confidence and ability | How enjoyment and confidence be influenced by the feeling of capability when completing a learning task. | A student feels satisfied after successfully completing an assignment. |- | Personal interest | How a person's interest in a topic increase curiosity and encourage them to learn more. | A student enjoys playing the flute as they are interested in joining the concert band. |- |Perceived usefulness |How realising the usefulness of knowledge make can make learning more meaningful. |A player enjoys learning about the different attacks in volleyball as they can apply that knowledge during the game. |- |Teaching methods |How the different methods such as activities and approches can encourage students willingness to participate. |A student enjoys the chemistry experiment after learning the ways chemicals can mix together. |- |Learning environment |How the atmosphere of the room, teachers support, and student interactions can increase enjoyment. |A student feels comfortable sharing ideas as the teacher encourages participation. |} === Research on Enjoyment in Learning === Research shows how personal characteristics and environmental factors influence student's enjoyment and their learning experiences. When a person has an interest in a subject or topic, they are more likely to learn more, pay attention, and retain the information. According to Harackiewicz et al. (2016), students who have an interest in a subject can increase their concentration, and their willingness to learn more. This shows that personal interest in a topic can lead to stronger understanding, higher participation, and more positive learning experiences. Enjoyment can also be influenced by the environment the person is in which can result in more retained information. According to Hernik and Jaworska (2018), students who attended a class that had a more enjoyable atmosphere illustrated a better recall and happiness. Additionally, research also shows that different teaching approaches and social activities can result in higher participation rates among students (Bonney, 2015). This shows that the classroom conditions, teaching approaches and social aspects can greatly contribute to a higher retention rate, and a more enjoyable learning experience. Let's look at another scenario. {{Robelbox|title=Scenario|iconwidth=55px|theme=5}}<div style="{{Robelbox/pad}}"> [[File:Hands raised (7645867).jpg|thumb|226x226px|Figure 3. Students in the classroom raised their hands while the teacher picked a student.]] Linda is learning about the chemical compounds in a science laboratory, and she loves it. Their teacher breaks down each concept into small, manageable chunks and encourages students to participate. Poppy is learning maths in a computer lab, but she is not particularly interested in maths. The teacher explains the steps it takes to solve a problem and quickly instructs the students to answer math problems using the computers. However, Poppy got bored and ended up playing games for the remainder of the session. Which student do you think had a higher recall of the topic in the next lesson? <div>{{Robelbox/close}}{{Robelbox|width=75%|theme=4|iconwidth=55px|title=Test your knowledge!}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. Which of the following is an example of how a learning environment can affect a student's retention? |type="()"} + A student retains information more easily because the classroom was quiet. - QA student enjoys learning because they are interested in the subject. - A student feels confident because they understand he topic. </div> </quiz> </div>{{Robelbox/close}} == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bonney, K. M. (2015). Case study teaching method improves student performance and perceptions of learning gains. Journal of Microbiology & Biology Education, 16(1), 21–28. https://doi.org/10.1128/jmbe.v16i1.846 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Moghadasi, J., & Keikavoosi-Arani, L. (2023). Investigating the factors influencing students’ academic enthusiasm for a shift of paradigm among education managers shaping academic pedagogy. BMC Medical Education, 23(1), 480. PubMed Central. https://doi.org/10.1186/s12909-023-04453-4 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Sabaratnam, A. (2025). The impact of students’ self-confidence on their learning interest. International Journal of Science and Research Archive, 15(2), 1632–1641. https://doi.org/10.30574/ijsra.2025.15.2.1599 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] lih5l2e1sisux7fft1gqebo29jwl3kg 2844133 2844132 2026-11-03T08:33:26Z Diaz Chas 3109441 /* Enjoyment vs Fun */ spacing 2844133 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Enjoyment in learning can be influenced by several different factors. Factors such as confidence, and interest are personal characteristics that may affect how much a person enjoys learning (Sabaratnam, 2025). Other factors such as the usefulness of the knowledge, the atmosphere, and the teaching methods can also influence enjoyment (Moghadasi & Keikavoosi-Arani, 2023). These factors can influence a person's motivation, participation and their overall learning experience. The table below outlines some key factors that may influence enjoyment in learning and some examples on how they can affect an individual. {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- |Confidence and ability | How enjoyment and confidence be influenced by the feeling of capability when completing a learning task. | A student feels satisfied after successfully completing an assignment. |- | Personal interest | How a person's interest in a topic increase curiosity and encourage them to learn more. | A student enjoys playing the flute as they are interested in joining the concert band. |- |Perceived usefulness |How realising the usefulness of knowledge make can make learning more meaningful. |A player enjoys learning about the different attacks in volleyball as they can apply that knowledge during the game. |- |Teaching methods |How the different methods such as activities and approches can encourage students willingness to participate. |A student enjoys the chemistry experiment after learning the ways chemicals can mix together. |- |Learning environment |How the atmosphere of the room, teachers support, and student interactions can increase enjoyment. |A student feels comfortable sharing ideas as the teacher encourages participation. |} === Research on Enjoyment in Learning === Research shows how personal characteristics and environmental factors influence student's enjoyment and their learning experiences. When a person has an interest in a subject or topic, they are more likely to learn more, pay attention, and retain the information. According to Harackiewicz et al. (2016), students who have an interest in a subject can increase their concentration, and their willingness to learn more. This shows that personal interest in a topic can lead to stronger understanding, higher participation, and more positive learning experiences. Enjoyment can also be influenced by the environment the person is in which can result in more retained information. According to Hernik and Jaworska (2018), students who attended a class that had a more enjoyable atmosphere illustrated a better recall and happiness. Additionally, research also shows that different teaching approaches and social activities can result in higher participation rates among students (Bonney, 2015). This shows that the classroom conditions, teaching approaches and social aspects can greatly contribute to a higher retention rate, and a more enjoyable learning experience. Let's look at another scenario. {{Robelbox|title=Scenario|iconwidth=55px|theme=5}}<div style="{{Robelbox/pad}}"> [[File:Hands raised (7645867).jpg|thumb|226x226px|Figure 3. Students in the classroom raised their hands while the teacher picked a student.]] Linda is learning about the chemical compounds in a science laboratory, and she loves it. Their teacher breaks down each concept into small, manageable chunks and encourages students to participate. Poppy is learning maths in a computer lab, but she is not particularly interested in maths. The teacher explains the steps it takes to solve a problem and quickly instructs the students to answer math problems using the computers. However, Poppy got bored and ended up playing games for the remainder of the session. Which student do you think had a higher recall of the topic in the next lesson? <div>{{Robelbox/close}}{{Robelbox|width=75%|theme=4|iconwidth=55px|title=Test your knowledge!}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. Which of the following is an example of how a learning environment can affect a student's retention? |type="()"} + A student retains information more easily because the classroom was quiet. - QA student enjoys learning because they are interested in the subject. - A student feels confident because they understand he topic. </div> </quiz> </div>{{Robelbox/close}} == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bonney, K. M. (2015). Case study teaching method improves student performance and perceptions of learning gains. Journal of Microbiology & Biology Education, 16(1), 21–28. https://doi.org/10.1128/jmbe.v16i1.846 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Moghadasi, J., & Keikavoosi-Arani, L. (2023). Investigating the factors influencing students’ academic enthusiasm for a shift of paradigm among education managers shaping academic pedagogy. BMC Medical Education, 23(1), 480. PubMed Central. https://doi.org/10.1186/s12909-023-04453-4 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Sabaratnam, A. (2025). The impact of students’ self-confidence on their learning interest. International Journal of Science and Research Archive, 15(2), 1632–1641. https://doi.org/10.30574/ijsra.2025.15.2.1599 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] 4vcywmxv9ngp03h5e6sgvzl92j9nmrj 2844134 2844133 2026-11-03T08:33:47Z Diaz Chas 3109441 /* Research on Enjoyment in Learning */ spacing 2844134 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Enjoyment in learning can be influenced by several different factors. Factors such as confidence, and interest are personal characteristics that may affect how much a person enjoys learning (Sabaratnam, 2025). Other factors such as the usefulness of the knowledge, the atmosphere, and the teaching methods can also influence enjoyment (Moghadasi & Keikavoosi-Arani, 2023). These factors can influence a person's motivation, participation and their overall learning experience. The table below outlines some key factors that may influence enjoyment in learning and some examples on how they can affect an individual. {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- |Confidence and ability | How enjoyment and confidence be influenced by the feeling of capability when completing a learning task. | A student feels satisfied after successfully completing an assignment. |- | Personal interest | How a person's interest in a topic increase curiosity and encourage them to learn more. | A student enjoys playing the flute as they are interested in joining the concert band. |- |Perceived usefulness |How realising the usefulness of knowledge make can make learning more meaningful. |A player enjoys learning about the different attacks in volleyball as they can apply that knowledge during the game. |- |Teaching methods |How the different methods such as activities and approches can encourage students willingness to participate. |A student enjoys the chemistry experiment after learning the ways chemicals can mix together. |- |Learning environment |How the atmosphere of the room, teachers support, and student interactions can increase enjoyment. |A student feels comfortable sharing ideas as the teacher encourages participation. |} === Research on Enjoyment in Learning === Research shows how personal characteristics and environmental factors influence student's enjoyment and their learning experiences. When a person has an interest in a subject or topic, they are more likely to learn more, pay attention, and retain the information. According to Harackiewicz et al. (2016), students who have an interest in a subject can increase their concentration, and their willingness to learn more. This shows that personal interest in a topic can lead to stronger understanding, higher participation, and more positive learning experiences. Enjoyment can also be influenced by the environment the person is in which can result in more retained information. According to Hernik and Jaworska (2018), students who attended a class that had a more enjoyable atmosphere illustrated a better recall and happiness. Additionally, research also shows that different teaching approaches and social activities can result in higher participation rates among students (Bonney, 2015). This shows that the classroom conditions, teaching approaches and social aspects can greatly contribute to a higher retention rate, and a more enjoyable learning experience. Let's look at another scenario. {{Robelbox|title=Scenario|iconwidth=55px|theme=5}}<div style="{{Robelbox/pad}}"> [[File:Hands raised (7645867).jpg|thumb|226x226px|Figure 3. Students in the classroom raised their hands while the teacher picked a student.]] Linda is learning about the chemical compounds in a science laboratory, and she loves it. Their teacher breaks down each concept into small, manageable chunks and encourages students to participate. Poppy is learning maths in a computer lab, but she is not particularly interested in maths. The teacher explains the steps it takes to solve a problem and quickly instructs the students to answer math problems using the computers. However, Poppy got bored and ended up playing games for the remainder of the session. Which student do you think had a higher recall of the topic in the next lesson? <div>{{Robelbox/close}}{{Robelbox|width=75%|theme=4|iconwidth=55px|title=Test your knowledge!}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. Which of the following is an example of how a learning environment can affect a student's retention? |type="()"} + A student retains information more easily because the classroom was quiet. - QA student enjoys learning because they are interested in the subject. - A student feels confident because they understand he topic. </div> </quiz> </div>{{Robelbox/close}} == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bonney, K. M. (2015). Case study teaching method improves student performance and perceptions of learning gains. Journal of Microbiology & Biology Education, 16(1), 21–28. https://doi.org/10.1128/jmbe.v16i1.846 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Moghadasi, J., & Keikavoosi-Arani, L. (2023). Investigating the factors influencing students’ academic enthusiasm for a shift of paradigm among education managers shaping academic pedagogy. BMC Medical Education, 23(1), 480. PubMed Central. https://doi.org/10.1186/s12909-023-04453-4 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Sabaratnam, A. (2025). The impact of students’ self-confidence on their learning interest. International Journal of Science and Research Archive, 15(2), 1632–1641. https://doi.org/10.30574/ijsra.2025.15.2.1599 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] rglbjptd7qtwbhu0bf8x079l8t2dvqc 2844135 2844134 2026-11-03T08:34:33Z Diaz Chas 3109441 /* Research on Enjoyment in Learning */ testing formatting 2844135 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Enjoyment in learning can be influenced by several different factors. Factors such as confidence, and interest are personal characteristics that may affect how much a person enjoys learning (Sabaratnam, 2025). Other factors such as the usefulness of the knowledge, the atmosphere, and the teaching methods can also influence enjoyment (Moghadasi & Keikavoosi-Arani, 2023). These factors can influence a person's motivation, participation and their overall learning experience. The table below outlines some key factors that may influence enjoyment in learning and some examples on how they can affect an individual. {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- |Confidence and ability | How enjoyment and confidence be influenced by the feeling of capability when completing a learning task. | A student feels satisfied after successfully completing an assignment. |- | Personal interest | How a person's interest in a topic increase curiosity and encourage them to learn more. | A student enjoys playing the flute as they are interested in joining the concert band. |- |Perceived usefulness |How realising the usefulness of knowledge make can make learning more meaningful. |A player enjoys learning about the different attacks in volleyball as they can apply that knowledge during the game. |- |Teaching methods |How the different methods such as activities and approches can encourage students willingness to participate. |A student enjoys the chemistry experiment after learning the ways chemicals can mix together. |- |Learning environment |How the atmosphere of the room, teachers support, and student interactions can increase enjoyment. |A student feels comfortable sharing ideas as the teacher encourages participation. |} === Research on Enjoyment in Learning === Research shows how personal characteristics and environmental factors influence student's enjoyment and their learning experiences. When a person has an interest in a subject or topic, they are more likely to learn more, pay attention, and retain the information. According to Harackiewicz et al. (2016), students who have an interest in a subject can increase their concentration, and their willingness to learn more. This shows that personal interest in a topic can lead to stronger understanding, higher participation, and more positive learning experiences. Enjoyment can also be influenced by the environment the person is in which can result in more retained information. According to Hernik and Jaworska (2018), students who attended a class that had a more enjoyable atmosphere illustrated a better recall and happiness. Additionally, research also shows that different teaching approaches and social activities can result in higher participation rates among students (Bonney, 2015). This shows that the classroom conditions, teaching approaches and social aspects can greatly contribute to a higher retention rate, and a more enjoyable learning experience. Let's look at another scenario. {{Robelbox|title=Scenario|iconwidth=55px|theme=5}}<div style="{{Robelbox/pad}}"> [[File:Hands raised (7645867).jpg|thumb|226x226px|Figure 3. Students in the classroom raised their hands while the teacher picked a student.]] Linda is learning about the chemical compounds in a science laboratory, and she loves it. Their teacher breaks down each concept into small, manageable chunks and encourages students to participate. Poppy is learning maths in a computer lab, but she is not particularly interested in maths. The teacher explains the steps it takes to solve a problem and quickly instructs the students to answer math problems using the computers. However, Poppy got bored and ended up playing games for the remainder of the session. Which student do you think had a higher recall of the topic in the next lesson? <div>{{Robelbox/close}}{{Robelbox|width=75%|theme=4|iconwidth=55px|title=Test your knowledge!}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. Which of the following is an example of how a learning environment can affect a student's retention? |type="()"} + A student retains information more easily because the classroom was quiet. - QA student enjoys learning because they are interested in the subject. - A student feels confident because they understand he topic. </div> </quiz> </div> {{Robelbox/close}} stuff == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bonney, K. M. (2015). Case study teaching method improves student performance and perceptions of learning gains. Journal of Microbiology & Biology Education, 16(1), 21–28. https://doi.org/10.1128/jmbe.v16i1.846 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Moghadasi, J., & Keikavoosi-Arani, L. (2023). Investigating the factors influencing students’ academic enthusiasm for a shift of paradigm among education managers shaping academic pedagogy. BMC Medical Education, 23(1), 480. PubMed Central. https://doi.org/10.1186/s12909-023-04453-4 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Sabaratnam, A. (2025). The impact of students’ self-confidence on their learning interest. International Journal of Science and Research Archive, 15(2), 1632–1641. https://doi.org/10.30574/ijsra.2025.15.2.1599 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] c7pwc94ydm9xnujvxv6sondc50vsa0u 2844137 2844135 2026-11-03T08:36:06Z Diaz Chas 3109441 /* Research on Enjoyment in Learning */ testing format 2844137 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Enjoyment in learning can be influenced by several different factors. Factors such as confidence, and interest are personal characteristics that may affect how much a person enjoys learning (Sabaratnam, 2025). Other factors such as the usefulness of the knowledge, the atmosphere, and the teaching methods can also influence enjoyment (Moghadasi & Keikavoosi-Arani, 2023). These factors can influence a person's motivation, participation and their overall learning experience. The table below outlines some key factors that may influence enjoyment in learning and some examples on how they can affect an individual. {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- |Confidence and ability | How enjoyment and confidence be influenced by the feeling of capability when completing a learning task. | A student feels satisfied after successfully completing an assignment. |- | Personal interest | How a person's interest in a topic increase curiosity and encourage them to learn more. | A student enjoys playing the flute as they are interested in joining the concert band. |- |Perceived usefulness |How realising the usefulness of knowledge make can make learning more meaningful. |A player enjoys learning about the different attacks in volleyball as they can apply that knowledge during the game. |- |Teaching methods |How the different methods such as activities and approches can encourage students willingness to participate. |A student enjoys the chemistry experiment after learning the ways chemicals can mix together. |- |Learning environment |How the atmosphere of the room, teachers support, and student interactions can increase enjoyment. |A student feels comfortable sharing ideas as the teacher encourages participation. |} === Research on Enjoyment in Learning === Research shows how personal characteristics and environmental factors influence student's enjoyment and their learning experiences. When a person has an interest in a subject or topic, they are more likely to learn more, pay attention, and retain the information. According to Harackiewicz et al. (2016), students who have an interest in a subject can increase their concentration, and their willingness to learn more. This shows that personal interest in a topic can lead to stronger understanding, higher participation, and more positive learning experiences. Enjoyment can also be influenced by the environment the person is in which can result in more retained information. According to Hernik and Jaworska (2018), students who attended a class that had a more enjoyable atmosphere illustrated a better recall and happiness. Additionally, research also shows that different teaching approaches and social activities can result in higher participation rates among students (Bonney, 2015). This shows that the classroom conditions, teaching approaches and social aspects can greatly contribute to a higher retention rate, and a more enjoyable learning experience. Let's look at another scenario. {{Robelbox|title=Scenario|iconwidth=55px|theme=5}}<div style="{{Robelbox/pad}}"> [[File:Hands raised (7645867).jpg|thumb|226x226px|Figure 3. Students in the classroom raised their hands while the teacher picked a student.]] Linda is learning about the chemical compounds in a science laboratory, and she loves it. Their teacher breaks down each concept into small, manageable chunks and encourages students to participate. Poppy is learning maths in a computer lab, but she is not particularly interested in maths. The teacher explains the steps it takes to solve a problem and quickly instructs the students to answer math problems using the computers. However, Poppy got bored and ended up playing games for the remainder of the session. Which student do you think had a higher recall of the topic in the next lesson? <div> {{Robelbox/close}} stuff {{Robelbox|width=75%|theme=4|iconwidth=55px|title=Test your knowledge!}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. Which of the following is an example of how a learning environment can affect a student's retention? |type="()"} + A student retains information more easily because the classroom was quiet. - QA student enjoys learning because they are interested in the subject. - A student feels confident because they understand he topic. </div> </quiz> </div> {{Robelbox/close}} stuff == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bonney, K. M. (2015). Case study teaching method improves student performance and perceptions of learning gains. Journal of Microbiology & Biology Education, 16(1), 21–28. https://doi.org/10.1128/jmbe.v16i1.846 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Moghadasi, J., & Keikavoosi-Arani, L. (2023). Investigating the factors influencing students’ academic enthusiasm for a shift of paradigm among education managers shaping academic pedagogy. BMC Medical Education, 23(1), 480. PubMed Central. https://doi.org/10.1186/s12909-023-04453-4 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Sabaratnam, A. (2025). The impact of students’ self-confidence on their learning interest. International Journal of Science and Research Archive, 15(2), 1632–1641. https://doi.org/10.30574/ijsra.2025.15.2.1599 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] 71i0lwrxltujx601i46bqrlrkvpm1n5 2844138 2844137 2026-11-03T08:36:57Z Diaz Chas 3109441 /* Enjoyment, motivation, and engagement */ fixed formatting issue 2844138 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Enjoyment in learning can be influenced by several different factors. Factors such as confidence, and interest are personal characteristics that may affect how much a person enjoys learning (Sabaratnam, 2025). Other factors such as the usefulness of the knowledge, the atmosphere, and the teaching methods can also influence enjoyment (Moghadasi & Keikavoosi-Arani, 2023). These factors can influence a person's motivation, participation and their overall learning experience. The table below outlines some key factors that may influence enjoyment in learning and some examples on how they can affect an individual. {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- |Confidence and ability | How enjoyment and confidence be influenced by the feeling of capability when completing a learning task. | A student feels satisfied after successfully completing an assignment. |- | Personal interest | How a person's interest in a topic increase curiosity and encourage them to learn more. | A student enjoys playing the flute as they are interested in joining the concert band. |- |Perceived usefulness |How realising the usefulness of knowledge make can make learning more meaningful. |A player enjoys learning about the different attacks in volleyball as they can apply that knowledge during the game. |- |Teaching methods |How the different methods such as activities and approches can encourage students willingness to participate. |A student enjoys the chemistry experiment after learning the ways chemicals can mix together. |- |Learning environment |How the atmosphere of the room, teachers support, and student interactions can increase enjoyment. |A student feels comfortable sharing ideas as the teacher encourages participation. |} === Research on Enjoyment in Learning === Research shows how personal characteristics and environmental factors influence student's enjoyment and their learning experiences. When a person has an interest in a subject or topic, they are more likely to learn more, pay attention, and retain the information. According to Harackiewicz et al. (2016), students who have an interest in a subject can increase their concentration, and their willingness to learn more. This shows that personal interest in a topic can lead to stronger understanding, higher participation, and more positive learning experiences. Enjoyment can also be influenced by the environment the person is in which can result in more retained information. According to Hernik and Jaworska (2018), students who attended a class that had a more enjoyable atmosphere illustrated a better recall and happiness. Additionally, research also shows that different teaching approaches and social activities can result in higher participation rates among students (Bonney, 2015). This shows that the classroom conditions, teaching approaches and social aspects can greatly contribute to a higher retention rate, and a more enjoyable learning experience. Let's look at another scenario. {{Robelbox|title=Scenario|iconwidth=55px|theme=5}}<div style="{{Robelbox/pad}}"> [[File:Hands raised (7645867).jpg|thumb|226x226px|Figure 3. Students in the classroom raised their hands while the teacher picked a student.]] Linda is learning about the chemical compounds in a science laboratory, and she loves it. Their teacher breaks down each concept into small, manageable chunks and encourages students to participate. Poppy is learning maths in a computer lab, but she is not particularly interested in maths. The teacher explains the steps it takes to solve a problem and quickly instructs the students to answer math problems using the computers. However, Poppy got bored and ended up playing games for the remainder of the session. Which student do you think had a higher recall of the topic in the next lesson? <div> {{Robelbox/close}} {{Robelbox|width=75%|theme=4|iconwidth=55px|title=Test your knowledge!}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. Which of the following is an example of how a learning environment can affect a student's retention? |type="()"} + A student retains information more easily because the classroom was quiet. - QA student enjoys learning because they are interested in the subject. - A student feels confident because they understand he topic. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bonney, K. M. (2015). Case study teaching method improves student performance and perceptions of learning gains. Journal of Microbiology & Biology Education, 16(1), 21–28. https://doi.org/10.1128/jmbe.v16i1.846 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Moghadasi, J., & Keikavoosi-Arani, L. (2023). Investigating the factors influencing students’ academic enthusiasm for a shift of paradigm among education managers shaping academic pedagogy. BMC Medical Education, 23(1), 480. PubMed Central. https://doi.org/10.1186/s12909-023-04453-4 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Sabaratnam, A. (2025). The impact of students’ self-confidence on their learning interest. International Journal of Science and Research Archive, 15(2), 1632–1641. https://doi.org/10.30574/ijsra.2025.15.2.1599 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] envghuu8omdqop007kunqkiqpmsos72 2844139 2844138 2026-11-03T08:37:22Z Diaz Chas 3109441 /* Research on Enjoyment in Learning */ spacing 2844139 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Enjoyment in learning can be influenced by several different factors. Factors such as confidence, and interest are personal characteristics that may affect how much a person enjoys learning (Sabaratnam, 2025). Other factors such as the usefulness of the knowledge, the atmosphere, and the teaching methods can also influence enjoyment (Moghadasi & Keikavoosi-Arani, 2023). These factors can influence a person's motivation, participation and their overall learning experience. The table below outlines some key factors that may influence enjoyment in learning and some examples on how they can affect an individual. {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- |Confidence and ability | How enjoyment and confidence be influenced by the feeling of capability when completing a learning task. | A student feels satisfied after successfully completing an assignment. |- | Personal interest | How a person's interest in a topic increase curiosity and encourage them to learn more. | A student enjoys playing the flute as they are interested in joining the concert band. |- |Perceived usefulness |How realising the usefulness of knowledge make can make learning more meaningful. |A player enjoys learning about the different attacks in volleyball as they can apply that knowledge during the game. |- |Teaching methods |How the different methods such as activities and approches can encourage students willingness to participate. |A student enjoys the chemistry experiment after learning the ways chemicals can mix together. |- |Learning environment |How the atmosphere of the room, teachers support, and student interactions can increase enjoyment. |A student feels comfortable sharing ideas as the teacher encourages participation. |} === Research on Enjoyment in Learning === Research shows how personal characteristics and environmental factors influence student's enjoyment and their learning experiences. When a person has an interest in a subject or topic, they are more likely to learn more, pay attention, and retain the information. According to Harackiewicz et al. (2016), students who have an interest in a subject can increase their concentration, and their willingness to learn more. This shows that personal interest in a topic can lead to stronger understanding, higher participation, and more positive learning experiences. Enjoyment can also be influenced by the environment the person is in which can result in more retained information. According to Hernik and Jaworska (2018), students who attended a class that had a more enjoyable atmosphere illustrated a better recall and happiness. Additionally, research also shows that different teaching approaches and social activities can result in higher participation rates among students (Bonney, 2015). This shows that the classroom conditions, teaching approaches and social aspects can greatly contribute to a higher retention rate, and a more enjoyable learning experience. Let's look at another scenario. {{Robelbox|title=Scenario|iconwidth=55px|theme=5}}<div style="{{Robelbox/pad}}"> [[File:Hands raised (7645867).jpg|thumb|226x226px|Figure 3. Students in the classroom raised their hands while the teacher picked a student.]] Linda is learning about the chemical compounds in a science laboratory, and she loves it. Their teacher breaks down each concept into small, manageable chunks and encourages students to participate. Poppy is learning maths in a computer lab, but she is not particularly interested in maths. The teacher explains the steps it takes to solve a problem and quickly instructs the students to answer math problems using the computers. However, Poppy got bored and ended up playing games for the remainder of the session. Which student do you think had a higher recall of the topic in the next lesson? <div> {{Robelbox/close}} {{Robelbox|width=75%|theme=4|iconwidth=55px|title=Test your knowledge!}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. Which of the following is an example of how a learning environment can affect a student's retention? |type="()"} + A student retains information more easily because the classroom was quiet. - QA student enjoys learning because they are interested in the subject. - A student feels confident because they understand he topic. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bonney, K. M. (2015). Case study teaching method improves student performance and perceptions of learning gains. Journal of Microbiology & Biology Education, 16(1), 21–28. https://doi.org/10.1128/jmbe.v16i1.846 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Moghadasi, J., & Keikavoosi-Arani, L. (2023). Investigating the factors influencing students’ academic enthusiasm for a shift of paradigm among education managers shaping academic pedagogy. BMC Medical Education, 23(1), 480. PubMed Central. https://doi.org/10.1186/s12909-023-04453-4 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Sabaratnam, A. (2025). The impact of students’ self-confidence on their learning interest. International Journal of Science and Research Archive, 15(2), 1632–1641. https://doi.org/10.30574/ijsra.2025.15.2.1599 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] c1zcg7y6vbb2e0tmd7bjlipjj6m79cu 2844149 2844139 2026-11-03T09:15:57Z Diaz Chas 3109441 /* Enjoyment, motivation, and engagement */ 2844149 wikitext text/x-wiki {{title|Enjoyment and learning:<br>How does enjoyment influence learning?}} ==Overview== === What is enjoyment in learning? === Learning is incorporated into everyday life and is a part of the brain's conscious and subconscious processes (Kuldas et al., 2013). It plays an important role in allowing individuals to develop new skills, understanding, and knowledge. For example, an individual may consciously focus on reading a book about road rules, while also absorbing and interpreting the information they read. Enjoyment in learning refers to the positive feelings an individual experiences when gaining new information or participating in learning activities (Song, 2024). It may also influence knowledge retention as individuals who enjoys a topic may be more likely to pay attention and process the information they are learning which increases their ability to remember it. Let's consider the following scenario. {{Robelbox|theme=10|title=Scenario|iconwidth=55px}}<div style="{{Robelbox/pad}}"> [[File:Students studying 005.jpg|thumb|210x210px|'''Figure 1'''. Students studying in a learning environment.]] Two students have an upcoming chemistry exam. Sarah enjoys chemistry and finds the content interesting and fun, while Lily finds the subject boring and has difficulty focusing. Although they spend the same amount of time studying and prepping for the exam, Sarah may be more motivated to engage and learn more about the material. Could Sarah's enjoyment of learning ultimately contribute to a higher exam mark than Lily's? </div>  {{Robelbox/close}} === Understanding Enjoyment and Learning === Enjoyment plays an important role in learning, as the positive experience individuals have while learning may influence how they interact with it (Lucardie, 2014). For example, when individuals enjoy what they are learning, their motivation and willingness to lean may increase. This is where understanding the psychology behind it can help explain how emotions, such as enjoyment and boredom, affect the way people learn. Psychological theories and research can also help explain how enjoyment may influence a person's motivation and willingness to learn (Bandhu et al., 2024). By understanding these relationships, we can gain a better understanding of whether enjoyment in learning makes it easier to focus, stay motivated, increase retention, and achieve better academic results. {{Robelbox|theme=5|title=Focus questions}} * What is enjoyment in the context of learning? * What factors contribute to enjoyment during learning? * How does enjoyment influence motivation and engagement? * How does enjoyment influence learning and academic outcomes? {{Robelbox/close}} == Enjoyment in learning == === Meaning of Enjoyment === Enjoyment in learning involves an individual experiences a positive feeling while participating or engaging in learning a new concept or topic (Mansfield, 2025). Some examples of positive feelings can be excitement in learning a new topic or feeling satisfied after gaining a better understanding on a concept an individual was interested in. Enjoyment can also occur when an individual gets informed about a topic that interests them prior to it getting discussed, which is also know as the anticipatory model (Studzinski Perotto, 2012). For example, when a teacher tells the students the topic that will be covered in the following class, and a student was interested in it or had prior knowledge they wanted to enhance, it can lead excitement for the future class. According to Wong et al. (2020), individuals may experience enjoyment when they gain a more in depth understanding about a subject they were interested in or may had prior knowledge on. This shows that interest and excitement can enhance the knowledge retained from learning. Understanding a topic that was deemed complex or difficult previously can create a sense of satisfaction, which is another sense of enjoyment. For example, an individual found a concept in biology difficult, and the following session the teacher explains it in a more simpler matter which made it easier to understand, which then leads to a sense of enjoyment and satisfaction. According to Piaget's theory of cognitive equilibrium, when an individual finally understands a concept they struggled with, they may experience a sense of satisfaction as the new understanding helped restore cognitive balance and addressed the confusion (Bormanaki & Khoshhal, 2017). However, personal enjoyment also plays an important role as people have different interests and experiences which can determine the level of enjoyment an individual may experience. === Enjoyment as an Emotion === Enjoyment is considered a positive emotion as it involves a pleasant feeling during a learning experience (Pekrun et al., 2011). Some examples are feelings of happiness, curiosity, and satisfaction that an individual may experience while learning. Enjoying a topic may also increase a person's interest and curiosity and encourage them to delve deeper. This may involve a neurotransmitter known as dopamine, as it plays an important role in the brains rewards and motivation sector that may encourage individuals to learn more and seek out rewarding experiences (Bromberg-Martin et al., 2010). Experiencing these positive feelings may also lead to individuals paying more attention and focusing on the learning materials. This also considers the concept of motivation, as individuals may be encouraged to continue learning due to the rewarding experience. For example, when a person enjoys what they are learning, they may be more willing to actively participate in activities, and learn more in the process. Enjoyment in learning can also encourage individuals to keep learning even if the content is difficult. However negative feelings such as frustration and boredom can also affect the way a person approach learning. Let's look at the scenario below. {{Robelbox|theme=10|iconwidth=55px|title=Scenario}}<div style="{{Robelbox/pad}}"> [[File:Professor and students in a university classroom in Tennessee.jpg|left|thumb|Figure 2. Teacher and students in a classroom during a lesson.]] Jared and Jordan are in the same Biology class. Jared loves butterflies and has always enjoyed learning about the life cycle, while Jordan only took the class to satisfy the science prerequisite, and he found science a bit difficult but he loves drawing. The teacher announces that the next session they will be discussing the life cycle of the butterfly. This made Jared super excited that he went home and started reading the textbook ahead of time. Jared on the other hand didn't pay much interest to what the teacher said and just headed home. the following day, when the teacher was discussing about the topic, Jared was so excited that he asked so many questions and actively participated in class. However, Jordan felt lost and frustrated because he didn't understand what the teacher was trying to explain so instead of paying attention, he started drawing in his notebook to pass time. Could Jordan's frustration and disinterest affect his knowledge retention about the life cycle in compared to Jared? <div>{{-}} {{Robelbox/close}} === Enjoyment vs Fun === Although Enjoyment and fun are closely related, they have different meanings and associations in the context of learning. Fun is usually associated with entertainment such as games or shopping that make people happy (Reis et al., 2016). Enjoyment in learning involve satisfying curiosity or gaining a new knowledge that releases a sense of satisfaction. For example, a student is enjoying a difficult maths problem because they feel satisfied after finally understanding how it is solved. This shows that learning does not always have to be entertaining to be enjoyable, as people can still enjoy activities that require effort and concentration. {{Robelbox|title=Test your knowledge!|theme=4|width=75%|iconwidth=55px}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. What role does dopamine play when a person experiences enjoyment while learning? |type="()"} + It helps regulate motivation and the brain's reward system. - It prevents people from feeling frustrated. - It stops people from becoming distracted. - It controls how much information a person can remember. </div> </quiz> </div> {{Robelbox/close}} == Factors that influence enjoyment in learning == Enjoyment in learning can be influenced by several different factors. Factors such as confidence, and interest are personal characteristics that may affect how much a person enjoys learning (Sabaratnam, 2025). Other factors such as the usefulness of the knowledge, the atmosphere, and the teaching methods can also influence enjoyment (Moghadasi & Keikavoosi-Arani, 2023). These factors can influence a person's motivation, participation and their overall learning experience. The table below outlines some key factors that may influence enjoyment in learning and some examples on how they can affect an individual. {| class="wikitable" style="table-layout: fixed; width: 100%; word-wrap: break-word;" ! style="width: 15%;" | Factors ! style="width: 50%;" | Explanation ! style="width: 35%;" | Example |- |Confidence and ability | How enjoyment and confidence be influenced by the feeling of capability when completing a learning task. | A student feels satisfied after successfully completing an assignment. |- | Personal interest | How a person's interest in a topic increase curiosity and encourage them to learn more. | A student enjoys playing the flute as they are interested in joining the concert band. |- |Perceived usefulness |How realising the usefulness of knowledge make can make learning more meaningful. |A player enjoys learning about the different attacks in volleyball as they can apply that knowledge during the game. |- |Teaching methods |How the different methods such as activities and approches can encourage students willingness to participate. |A student enjoys the chemistry experiment after learning the ways chemicals can mix together. |- |Learning environment |How the atmosphere of the room, teachers support, and student interactions can increase enjoyment. |A student feels comfortable sharing ideas as the teacher encourages participation. |} === Research on Enjoyment in Learning === Research shows how personal characteristics and environmental factors influence student's enjoyment and their learning experiences. When a person has an interest in a subject or topic, they are more likely to learn more, pay attention, and retain the information. According to Harackiewicz et al. (2016), students who have an interest in a subject can increase their concentration, and their willingness to learn more. This shows that personal interest in a topic can lead to stronger understanding, higher participation, and more positive learning experiences. Enjoyment can also be influenced by the environment the person is in which can result in more retained information. According to Hernik and Jaworska (2018), students who attended a class that had a more enjoyable atmosphere illustrated a better recall and happiness. Additionally, research also shows that different teaching approaches and social activities can result in higher participation rates among students (Bonney, 2015). This shows that the classroom conditions, teaching approaches and social aspects can greatly contribute to a higher retention rate, and a more enjoyable learning experience. Let's look at another scenario. {{Robelbox|title=Scenario|iconwidth=55px|theme=5}}<div style="{{Robelbox/pad}}"> [[File:Hands raised (7645867).jpg|thumb|226x226px|Figure 3. Students in the classroom raised their hands while the teacher picked a student.]] Linda is learning about the chemical compounds in a science laboratory, and she loves it. Their teacher breaks down each concept into small, manageable chunks and encourages students to participate. Poppy is learning maths in a computer lab, but she is not particularly interested in maths. The teacher explains the steps it takes to solve a problem and quickly instructs the students to answer math problems using the computers. However, Poppy got bored and ended up playing games for the remainder of the session. Which student do you think had a higher recall of the topic in the next lesson? <div> {{Robelbox/close}} {{Robelbox|width=75%|theme=4|iconwidth=55px|title=Test your knowledge!}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. Which of the following is an example of how a learning environment can affect a student's retention? |type="()"} + A student retains information more easily because the classroom was quiet. - QA student enjoys learning because they are interested in the subject. - A student feels confident because they understand he topic. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment, motivation, and engagement == * How enjoyment can affect student's motivation to learn. * How can enjoyment affect how engaged or involved students are in learning. * How enjoyment can affect the amount of effort students put into their work - (Zheng et al., 2026) * Whether enjoying learning can make students more likely to push through difficult or challenging content. * Find research about relationship between '''enjoyment, motivation, and engagement.''' * {{Robelbox|theme=4|title=Test yourself!|width=75%}}<div style="{{Robelbox/pad}}"> <quiz display="simple"> 1. How might enjoyment influence a student's engagement in learning? |type="()"} + It may encourage them to become more involved in learning activities. - It always guarantees higher marks. - It removes the need to put effort into learning. - It prevents students from finding tasks difficult. </div> </quiz> </div> {{Robelbox/close}} == Enjoyment and learning outcomes == Include an introductory paragraph before branching into sub-sections === Cognitive processes === * How enjoyment can affect [[wikipedia:Attention|attention]] and focus while learning - (Liu et al., 2025) * How enjoyment can affect memory and remembering information - (Tyng et al., 2017) * How enjoyment can affect the way students process or understand information. * Find research about '''enjoyment and cognitive processes during learning'''. === Academic achievement === * Whether enjoyment is related to better grades or academic performance - (Kang & Wu, 2022). * Whether students who enjoy learning are more likely to do well academically. * Whether enjoyment can lead to better results, or if doing well can also make learning more enjoyable. * Find research about the relationship between '''enjoyment and academic achievement.''' --- Add a quiz or a learning figure == Conclusion == * Enjoyment may play an important role in how students experience and approach learning. * Enjoyment may influence different parts of learning such as motivation, engagement, coginitve processes, and academic achievement. * Different factors may affect how much students enjoy learning such as their confidence, interest, and learning environment. * Overall, understanding enjoyment could help how positive emotions can support learning. {{Robelbox|theme=10|title=Take home message}} <div style="{{Robelbox/pad}}"> Finding ways to make learning more enjoyable could help students stay motivated and engaged with what they are learning. </div>  {{Robelbox/close}} == See also == * [[wikipedia:Psychology_of_learning|Psychology of learning]] (Wikipedia) * [[Motivation and emotion/Book/2021/Student engagement and learning|Student engagement and learning]] (Book chapter, 2021) == References == {{Hanging indent|1= Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. Acta Psychologica, 244(1), 1–16. https://doi.org/https://doi.org/10.1016/j.actpsy.2024.104177 Bonney, K. M. (2015). Case study teaching method improves student performance and perceptions of learning gains. Journal of Microbiology & Biology Education, 16(1), 21–28. https://doi.org/10.1128/jmbe.v16i1.846 Bormanaki, H. B., & Khoshhal, Y. (2017). The role of equilibration in Piaget’s theory of cognitive development and its implication for receptive skills: A theoretical study. Journal of Language Teaching and Research, 8(5), 996–1005. https://doi.org/10.17507/jltr.0805.22 Bromberg-Martin, E. S., Matsumoto, M., & Hikosaka, O. (2010). Dopamine in motivational control: Rewarding, aversive, and alerting. Neuron, 68(5), 815–834. https://doi.org/10.1016/j.neuron.2010.11.022 Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542 Hernik, J., & Jaworska, E. (2018). The effect of enjoyment on learning. INTED2018 Proceedings. https://doi.org/10.21125/inted.2018.1087 Kang, X., & Wu, Y. (2022). Academic enjoyment, behavioral engagement, self-concept, organizational strategy and achievement in EFL setting: A multiple mediation analysis. PLOS ONE, 17(4), e0267405. https://doi.org/10.1371/journal.pone.0267405 Kuldas, S., Ismail, H. N., Hashim, S., & Bakar, Z. A. (2013). Unconscious learning processes: Mental integration of verbal and pictorial instructional materials. SpringerPlus, 2(1). https://doi.org/10.1186/2193-1801-2-105 Liu, H., Jiang, Y., Nie, H., & Zhou, X. (2025). Unveiling the effects of enjoyment and hope on students’ English learning effort. Scientific Reports, 15(1). https://doi.org/10.1038/s41598-025-11904-z Lucardie, D. (2014). The impact of fun and enjoyment on adult’s learning. The Fourth International Conference on Adult Education, Romania 2014, 142, 439–446. https://doi.org/10.1016/j.sbspro.2014.07.696 Mansfield, C. (2025). ‘We have learning objectives, not enjoyment objectives’: The role of enjoyment in life sciences teaching and learning. Journal of Learning Development in Higher Education, (35). https://doi.org/10.47408/jldhe.vi35.1318 Moghadasi, J., & Keikavoosi-Arani, L. (2023). Investigating the factors influencing students’ academic enthusiasm for a shift of paradigm among education managers shaping academic pedagogy. BMC Medical Education, 23(1), 480. PubMed Central. https://doi.org/10.1186/s12909-023-04453-4 Pekrun, R., Goetz, T., Frenzel, A. C., Barchfeld, P., & Perry, R. P. (2011). Measuring emotions in students’ learning and performance: The achievement emotions questionnaire (AEQ). Contemporary Educational Psychology, 36(1), 36–48. https://doi.org/10.1016/j.cedpsych.2010.10.002 Reis, H. T., O’Keefe, S. D., & Lane, R. D. (2016). Fun is more fun when others are involved. The Journal of Positive Psychology, 12(6), 547–557. https://doi.org/10.1080/17439760.2016.1221123 Sabaratnam, A. (2025). The impact of students’ self-confidence on their learning interest. International Journal of Science and Research Archive, 15(2), 1632–1641. https://doi.org/10.30574/ijsra.2025.15.2.1599 Song, Y. (2024). Assessing the interactions between learning enjoyment, motivation, burnout, and grit in EFL students: A mixed-methods approach. BMC Psychology, 12(1). https://doi.org/10.1186/s40359-024-02303-6 Studzinski Perotto, F. (2012). Anticipatory learning mechanisms. Encyclopedia of the Sciences of Learning, 266–269. https://doi.org/10.1007/978-1-4419-1428-6_1873 Tyng, C. M., Amin, H. U., Saad, M. N. M., & Malik, A. S. (2017). The influences of emotion on learning and memory. Frontiers in Psychology, 8, 1454. PubMed Central. https://doi.org/10.3389/fpsyg.2017.01454 Wong, L.-H., Chan, T.-W., Chen, W., Looi, C.-K., Chen, Z.-H., Liao, C. C. Y., King, R. B., & Wong, S. L. (2020). IDC theory: Interest and the interest loop. Research and Practice in Technology Enhanced Learning, 15(1), Article 3. https://doi.org/10.1186/s41039-020-0123-2 Zheng, H., Ye, Z., Bai, X., Shen, Y., & Cui, G. (2026). The effect of enjoyment on the achievement of learning goals in college students’ online classes: A moderated mediation model. Humanities and Social Sciences Communications, 13(1). https://doi.org/10.1057/s41599-026-06563-y }} == External links == * [https://www.academy.vic.gov.au/resource-collection/student-engagement Student engagement] (Victorian Academy of Teaching and Leadership) * [https://www.hunschool.org/admission/resources/blog/the-impact-of-enjoyment-on-learning-retention The impact of enjoyment on learning retention] (The Hun School of Princeton) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Joy]] [[Category:Motivation and emotion/Book/Learning]] m18oupyjjipw1ns4k4bbdy8eb8rpc8c 1 331470 2844072 2837204 2026-11-03T00:10:12Z Jtneill 10242 Book chapter feedback 2844072 wikitext text/x-wiki == See Also Addition == Hi, I read through your chapter and it looks like you are well on your way to writing a fabulous book chapter! I wanted to mention that you should add the book chapter [[Motivation and emotion/Book/2026/Emotion dysregulation|Emotion dysregulation]] in your 'see also' section because that chapter will be able to provide further information on emotion dysregulation as a whole. Good Luck with the book chapter! [[User:U3284040|U3284040]] ([[User talk:U3284040|discuss]] • [[Special:Contributions/U3284040|contribs]]) 01:35, 25 August 2026 (UTC) :Hi, :Thank you for your suggestion. :I have included book chapter 'Emotion dysregulation' in my see also section. [[User:P U3270518|P U3270518]] ([[User talk:P U3270518|discuss]] • [[Special:Contributions/P U3270518|contribs]]) 03:19, 27 August 2026 (UTC) == GenAI Acknowledgement Statement == I have used the following Generative Artificial Intelligence (GenAI) tools/services in the preparation of this assessment: GenAI service name and version: Microsoft copilot and Studiosity plus writing feedback '''(See edit history/summary for link to the chat of GenAI)''' Ways this GenAI service was used: • for initial ideation in response to the topic • used to understand book chapter question • creation of book chapter outline and brainstorm ideas • used to paraphrase few section • for finding synonyms of few words • used to ask how much knowledge I should have about the topic to answer the question • helped in summarising few articles and find articles • used to ask question on how I can approach this question • summarising difficult texts in simple language • creation of initial draft • finding articles related to the topic • assisting in structuring my thinking • used in checking if reference is in APA 7th format • checking grammar and spelling I confirm that I have not used GenAI in the preparation of this assessment for any purpose other than what I have acknowledged above, and I have cited and referenced any GenAI content in my assessment submission, applying the relevant referencing style. I understand that providing false or misleading information in this GenAI Acknowledgement Statement may constitute a breach of the University of Canberra (Student Conduct) Rules 2023. --[[User:P U3270518|P U3270518]] ([[User talk:P U3270518|discuss]] • [[Special:Contributions/P U3270518|contribs]]) 06:16, 8 October 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> <!-- Heading structure --> <!-- 2-level --> # Excellent – Well developed [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]]. Meaningful headings clearly relate directly to the core topic. <!-- Alignment with focus questions --> # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement in making the wording more similar <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section <!-- Description --> # A clear description of the problem/topic is planned or presented <!-- Style --> # Use present, rather than future, tense <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Contrast with and connect to negative emotion dysregulation # Perhaps consider how people '''cope''' with position emotion; maladaptive coping is dysregulation # Provide more detailed edit summaries <!-- Scope --> # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title <!-- Writing style --> # The writing style is clear and easy to follow <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- GenAI ---> # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Promising use of scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) # Place each quiz question in the most relevant section <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Excellent ## Use [[w:Letter case#Sentence casing|sentence casing]] |9= <!-- User page --> # Excellent <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:52, 1 September 2026 (UTC) == Edit Suggestions == Great work. '''Scenario.''' When I read your scenario about Grace, I noticed that you change tense a couple times, so I went back over it and recommend a few tweaks: # Grammar: You need a THE in 'for next several days' # Tense: Consider re-writing 'Her close friends tried to slow her down, but she ignored them' - to something like 'Her friends try to slow her down'. # Grammar: 'if something is not right' doesn't flow right, try something like 'whether something is wrong' # Tense: 'Why do her emotions become so intense' - try ' Why have' or 'why did' her '''Quiz:''' Recommend you review Quiz 1 and simplify the wording. When you compare that quiz to the other two, it is far more complex. '''Focus Questions and Headings:''' Double check you have said what you want to say and the subheadings are all aligned. e.g. Your Q3: How does positive emotion dysregulation affect psychological functioning and wellbeing? States that it will look at wellbeing, but when you read the chapter sub section, you don't mention wellbeing again until the quiz and the conclusion. Perhaps wellbeing is a hangover from an earlier draft. If you don't need it, remove it from focus question. '''Take home messages.''' Perhaps remove the em dash lines and either write it as a concise paragraph, or use the same number format that you used in your focus question box. [[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 04:11, 9 October 2026 (UTC) == Suggestion to integrate BAS and dopaminergic pathways == Hi, great work on this chapter! Such an interesting topic. Under 'Biological and temperamental factors', you might like to consider mentioning the Behavioural Activation System (BAS) and dopaminergic reward sensitivity (e.g., Carver & Johnson, 2009; Depue & Collins, 1999; Reeve, 2024, Ch. 3). Explaining that BAS hyper-activity (via pathways in the ventral tegmental are and nucleus accumbens) drives rash impulsivity and positive urgency would provide a strong neurobiological foundation alongside Linehan's Biosocial model. Thanks for the informative read! [[User:U3330981|U3330981]] ([[User talk:U3330981|discuss]] • [[Special:Contributions/U3330981|contribs]]) 06:21, 9 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged, however links to conversations (e.g., https://copilot.cloud.microsoft/chat/conversation/87b9977b-fde7-479e-89c8-2abea019f8e3?fromcode=cmmqbg1y3ah&es=UnauthClick) were not publicly accessible <!-- Overall – Citations --> # Excellent use of academic, peer-reviewed citations to support claims <!-- Overall – Word count --> # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FPositive_emotion_dysregulation&diff=2844071&oldid=2837316 these copy edits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Reasonably good scenario or case study; could more compellingly illustrate PED # Use an example that a wider audience can relate to; less than 5% of the world population are university students <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are very good # Perhaps also consider how PED can be regulated/managed/treated |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Reduce general theoretical background (e.g., emotion regulation). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question (PED)). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used <!-- Theory – Examples --> # Excellent use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Claims are well referenced <!-- Research – Critical thinking --> # Excellent [[w:Critical thinking|critical thinking]] about relevant research is evident |5= <!-- Integration comments... --> # Excellent integration between the most relevant theory(ies) and the best research |6= <!-- Conclusion comments... --> # Very good summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Somewhat clear take-home message(s) # Add more practical, take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections <!-- Style – Grammar --> # Grammar and spelling are excellent ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Figures --> ## Figures ### Very well captioned # There is a spelling error in Figure 2 (receives) <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Style – Tables --> ## Table(s) ### Very well captioned <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses <!-- Style – References --> ## References use very good APA style: ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] |8= <!-- Learning features comments... --> # Excellent of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent use of figure(s) # Excellent use of table(s) # Excellent use of feature box(es) # Reasonably good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Excellent use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent use of the [[{{PAGENAME}}#See also|See also]] section # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent use of the [[{{PAGENAME}}#External links|External links]] section # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~60 logged, useful contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # Thanks very much for your extensive contributions }} ikmdem021gdghj7lj7eiceclfs441f1 2 331498 2844007 2842895 2026-11-02T19:39:19Z Jtneill 10242 /* Social Contributions */ {{ic|Use a numbered list as taught in Tutorial 2}} 2844007 wikitext text/x-wiki == About Me == [[Motivation and emotion/Book/2026/Technology-based pain management|Motivation and emotion/Book/2026/Technology-based pain management - Wikiversity]] My name is Chelsea Schofield. I am a university student at Canberra University, currently completing a Bachelor of Science in Psychology with a major in Justice Studies. I have a 10 year history in the community sector, holding Certificates III and IV in Community Services. I am passionate about psychology, mental health advocacy, community advocacy, social work, nursing, and many other care-related fields. Over the years, I have worked in a variety of areas in the community sector, and volunteered in mental health support and drug awareness programs. I am committed to promoting wellbeing and making a positive impact within my community. === Education === * Bachelor of Science in Psychology, majoring in Justice Studies – Current * Certificate IV in Community Services * Certificate III in Community Services === interests and Focus Areas === * Mental health advocacy * Community advocacy * Social work * Nursing and healthcare * Justice studies and legal frameworks * Community services === Volunteer Experience === * Mental health support programs * Mental Health and Drug Awareness Campaign === Goals and Aspirations === * Contribute to mental health and community support programs * Develop expertise in social work and justice-related fields * Share knowledge and resources to support students and community members * Advocate for positive change within the care industry === Social contributions === {{ic|Use a numbered list as taught in Tutorial 2}} https://uclearn.canberra.edu.au/courses/20143/discussion_topics/455261?entry_id=810082 https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456345?entry_id=810128 [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457902 Topic: The Potential Future of Pain Technology, Would You consider A Brain Chip ?Motivation and emotion/Book/2026/Technology-based pain management - Wikiversity] [[Motivation and emotion/Book/2025/Positive criminology and motivation#Social theories]] se03d9x1yibghma1j7rmq74b1acxdor 1 331509 2844084 2843690 2026-11-03T02:57:51Z Jtneill 10242 2844084 wikitext text/x-wiki == Overview scenario == Hey there! Super interesting topic you have chosen, as well as an awesome start to hook your readers in. The only additional suggestion I would make is to place your scenario into a feature box, to make it really stand out. Other than that, great start. [[User:LMM26|LMM26]] ([[User talk:LMM26|discuss]] • [[Special:Contributions/LMM26|contribs]]) 13:59, 25 August 2026 (UTC) == User page == Hi there, Interesting topic. I have a small suggestion regarding your page. It might be better to create a separate user page for your personal information and social contributions, rather than including it on the same page as your book chapter. This would help keep the book chapter page focused and organised. Thanks and all the best.--[[User:P U3270518|P U3270518]] ([[User talk:P U3270518|discuss]] • [[Special:Contributions/P U3270518|contribs]]) 11:26, 3 September 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] # Remove bold from headings (use default heading styles) <!-- Heading structure --> <!-- 1-level --> # Basic [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – would benefit from further development (e.g., consider using subheadings) # The stand-alone research section isn't needed; instead, integrate research into the other sections # The emphasis appears to be on social aspects, but also consider other theoretical (e.g., physiological, emotional, cognitive) <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure |3= <!-- Overview--> # Good <!-- Scenario --> # A basic scenario is presented. The planned scenario is somewhat abstract and lacking in viscerality. Consider providng a more user-friendly scenario. # I moved an image into the feature box to help attract reader interest <!-- Description --> # A promising description of the problem/topic is planned or presented # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. <!-- Style --> # Use present, rather than future, tense <!-- Focus questions --> # Develop closer alignment between the sub-title and the focus questions |4= <!-- Key points--> <!-- Overall --> # Promising development # Highlight the most relevant theories and synthesise the best research on the topic # Focus on providing an integrative review of the most relevant theories and research on the topic, rather than describing individual study after individual study # The proposed chapter overly emphasises research and insufficiently emphasises theory # Provide more detailed edit summaries <!-- Scope --> # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title <!-- Writing style --> # The writing style is generally clear but could be simplified or made more concise <!-- Citations --> # Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Conclusion --> # Conclusion is underway # Overly focused on research; insufficiently focused on thoery # At risk of being overly detailed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Relevant figure(s) are presented and captioned <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also link to [[Motivation and emotion/Book|book chapters]] <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies # Make the relevance of the scenario to the topic more clear # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Consider including quiz question(s) about the take-home messages (instead of the key points feature boxes) <!-- Tables --> # Excellent use of [[Motivation and emotion/Wikiversity/Tables|table(s)]] # Add table captions |7= <!-- References --> <!-- Overall --> # Very good <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] |8= <!-- Resources --> <!-- See also --> # See also ## Insufficient ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use internal link style as shown in Tutorial 2 ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## Move academic link into References and cite <!-- External links --> # External links ## Basic ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link |9= <!-- User page --> # Used effectively <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:17, 10 September 2026 (UTC) == Feedback == Hey, I think you chose a really interesting topic and could have taken it in a lot of directions such as it being from the motivation of the children themselves, their parents, educators, etc. I thought the opening scenario was a great way of introducing outdoor play and showing that it can involve autonomy, social interaction, emotional challenges and problem solving rather than simply physical activity. I also liked that you acknowledge early on that emotional well being is not a single outcome and can include emotional dysregulation, social, emotional competence and positive/negative affect. One suggestion I would make for the Overview is to condense some of the research detail and move it into the later Research section. For example, the Overview currently includes specific sample sizes, study methods and mediation analysis. Replacing some of this detail with a shorter explanation of why outdoor play may influence emotional well being could make the Overview more accessible to a general reader. [[User:Reillyu3280706|Reillyu3280706]] ([[User talk:Reillyu3280706|discuss]] • [[Special:Contributions/Reillyu3280706|contribs]]) 08:52, 14 September 2026 (UTC) == Feedback == Hi there, I think your topic is really interesting, and I really enjoyed how your conclusion brought everything together. I also liked your section on working memory and your quizzes throughout to test understanding. I have made some comments in italic square brackets throughout for your consideration where you may need to insert a reference or where information is repetitive, and made grammatical corrections throughout. I also noticed that you link the definition to emotion regulation multiple times, you might want to consider only having the link the first time you mention it. I suggest that before submitting, you read it aloud to yourself to check grammar and clarity. I hope this was helpful! [[User:U3254168|U3254168]] ([[User talk:U3254168|discuss]] • [[Special:Contributions/U3254168|contribs]]) 02:10, 27 September 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a borderline sufficient chapter # A lot of work has been put in, but it is a scattergun which isn't well targetted at the topic; a lot of the content talks around the topic and some it is related to the topic # The main area(s) for potential improvement: #* address the topic instead of talking around the topic i.e., ''how'' does OP influence EWB? #* use the best psychological theories about the topic #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* more detailed review of the best psychological research about the topic (''how'' does OP influence CEB?); this chapter does not sufficiently narrow in to focus on the "how"; it talks around the topic #* quality of written expression #* integrate more strongly with other chapters by using embedded links to related topics #* possible overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI|genAI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words <!-- Overall - Edit summaries --> #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FOutdoor_play_and_children%27s_emotional_well-being&diff=2843689&oldid=2839134 these copy edits] |2= <!-- Overview – Comments... --> # Reasonably good <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Reasonably good scenario or case study; could be more compelling <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Insufficient explanation of the psychological problem or phenomenon # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions lack sufficient precision to the target topic # FQs 3, 4, and 5 are unnecessary; research should be integrated throughout # The focus on the chapter should more specifically on FQs 1 and 2 |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Insufficient use of relevant psychological theory about this topic # Expand on the cognitive and social theories # Also consider developmental psychological theories # Also consider environmental/evolutionary theories e.g., biophilia hypothesis, stress reduction theory, attention restoration theory <!-- Theory – Builds on --> # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Very good use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Examples --> # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Promising emphasis on research but problematic focus; overall, a basic review of ''relevant'' research # The main areas for improvement: ## Select more relevant research (i.e., that helps understand the theoretical mechanisms (the ''how'') involved in explaining the effect of children's outdoor play on EWB) ## Synthesise/integrate that research # Insufficient use of relevant systematic reviews and/or meta-analyses <!-- Research – Critical thinking --> # Reasonably good [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships |5= <!-- Integration comments... --> # Insufficient integration between the ''most relevant theory(ies)''' and the ''best research'' # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # A reasonably good summary of the chapter, but this is insufficient as a cohesive summary of the best psychological theory and research about the topic <!-- Conclusion – Key points --> # Key points are well summarised <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is reasonably good ## Reduce excessive use of bold font <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) <!-- Style – Copy editing --> # Copy editing ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] <!-- Style – Figures --> ## Figures ### Very well captioned ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Style – Tables --> ## Table(s) ### Very well captioned <!-- Style – Tables - Citations --> ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] <!-- Style – References --> ## References use reasonably good APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### More copy editing is needed to fix formatting inconsistencies |8= <!-- Learning features comments... --> # Very good use of learning features <!-- Learning features – Wikipedia embedded links --> # Promising use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent use of figure(s) # Excellent use of table(s) # Very good use of feature box(es) # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Good use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent use of the [[{{PAGENAME}}#See also|See also]] section <!-- Learning features – External links --> # Good use of the [[{{PAGENAME}}#External links|External links]] section ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order |9= <!-- Social contribution comments... --> # ~1 logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~1 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. }} bi9kv97uwh353mlbqolc16bkfh8mnrf 0 331583 2844023 2840877 2026-11-02T21:16:28Z Jtneill 10242 Copyediting 2844023 wikitext text/x-wiki {{title|Cognitive versus affective empathy:<br>What are the differences between cognitive and affective empathy and how do they contribute to prosociality?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Empathy and the brain.png|thumb|150px|'''Figure 1'''. Affective and cognitive empathy {{ic|Explain how this image relates to the scenario.}}]] '''Scenario''' Imagine a friend tells you they accidentally deleted the entirety of their important assignment. You might try to understand what they are thinking and feeling. At the same time, you may feel sadness, concern or anxiety in response to their experience. Both responses can be described as empathy, but they are not necessarily the same psychological process. {{RoundBoxBottom}} Empathy is a multidimensional psychological construct and is important in human interaction as it allows a person to understand and respond to the experience of another. Empathy can involve responding to another persons{{g}} psychological or emotional state, while maintaining a distinction between self and other. Often confused with sympathy, empathy is not simply defined as feeling sorry for someone, but involves gaining an understanding of someones situation and their psychological response (Maibom, 2020). Empathy is often viewed as putting yourself in another's shoes however, {{g}} research {{f}} shows that empathy involves different processes and adoptions of empathy. There are two main constructs of empathy, affective and cognitive {{f}}. Both types of empathy can influence how people interact with one another and can encourage prosocial behaviour through cooperating, supporting, comforting and helping others (Sanders et al., 2021). {{ic|So, what is the difference?}} {{RoundBoxTop|theme=3}} ;Focus questions * What is the difference between cognitive and affective empathy? * How does each form of empathy contribute to prosocial behaviour? * When can empathic responses fail to produce helpful behaviour? {{RoundBoxBottom}} ==Development of Empathy== Empathy is developed progressively throughout early childhood and adolescence as one develops emotional, cognitive and social skills {{f}}. it {{g}} is believed that empathy can develop through interactions between maturation, environmental influences and social experiences (GGSC, 2026). [[File:There's no crying in baseball! (4549295140) 2.jpg|thumb|Figure 2. A Child Comforting Another in Emotional Distress]] For most individuals, signs of affective empathy appear early in life {{f}}. Using foundations of emotional cognition, children will begin to cry at the sight of another infant crying {{f}}. This does not represent a full understanding of another's emotions however, it reflects early sensitivity to the emotional state of those around them. As the growth of children progresses, they eventually learn to distinguish the emotions of self and other to respond in the most appropriate way {{f}}. In a study addressing prosocial behaviour and empathy in relation to ageing, researchers found {{how}} that adults responded {{how}} strongly to empathy and were more likely to contribute helpful actions as a result. The study suggests that as people age they develop stronger empathetic abilities and display more frequent prosocial tendencies (Beadle et al., 2013). Cognitive empathy is developed later in life as is utilises more advance thinking skills. Children learn that other people will have their own individual beliefs, thoughts and perspectives that may differer from their own. According to Zephyra Merritt (2026), Perspective taking abilities begin to develop between 6-10 years of age, allowing children to understand that others might view situations differently from themselves. During this period of life, children become capable of seeing how another person might feel and adjusting their behaviour accordingly {{f}}. == Perspective Taking == Perspective taking is the ability to mentally put oneself in the position of another to accurately understand how they might view their situation. Perspective taking is closely related to empathy as it allows people to understand the experiences of others, but is not identical to empathy {{f}}. Perspective taking forms a cognitive foundation of empathy ands understanding, giving one the opportunity to have more effective social interactions and responses (Reniers et al., 2022). ==Cognitive Empathy== Cognitive empathy involves understanding another persons {{g}} psychological perspective without experiencing the same emotions {{f}}. It is the ability to recognise how another person may be thinking or feeling and considering a situation from another persons point of view {{f}}. Some signs of effective empathy include active listening, strong perspective taking skills, thoughtful questioning, awareness of social cues, open minded thinking and adaptable communication (z. Merritt, 2026). Cognitive empathy has relation {{aw}} to perspective taking, involving mental consideration of one's viewpoint {{f}}. Cognitive empathy plays an important role in social interaction and communication {{f}}. It enables one to interpret the beliefs, intentions and emotions of others, helping them respond to their situation appropriately. {{g}} some examples may include A manager noticing the silence of an employee and understand that this may be stress rather than disengagement. A teacher may recognise a poorly performing student as being anxious and not lazy. In cases such as these, cognitive empathy can help people to make the right informed, and compassionate decision (Price, 2023). The development of cognitive empathy is important in professions that require effective interpersonal relationships including education, healthcare and counselling. by understanding different view points, professionals can communicate effectively, manage conflict, and provide individually tailored support (Fink & de Rosnay, 2023). However, cognitive empathy is considered morally neutral as it can be used for both positive and negative purposes and is not inherently prosocial {{f}}. while it can often promote understanding and cooperating, it can also be noticed in manipulative individuals who understand others' emotions without caring for their wellbeing. In such cases, understanding someones psychological state can be used for self interest, deception or exploitation rather than helping (Reniers et al., 2022). Unaccompanied by affective empathy, cognitive empathy can not solely be responsible for morally correct behaviour. == Affective Empathy == Affective empathy is the capacity to emotionally respond to and share the feelings of another person {{f}}. Affective empathy unlike cognitive empathy, involves experiencing and emotional reaction that mirrors or embodies the emotions of another person {{f}}. Affective empathy plays a crucial role in forming and maintaining social relationships {{f}}. when others' witness someone experiencing joy, fear, distress or sadness, they may experience a similar emotional reaction {{f}}. Emotional resonance such as this, helps strengthen relationships and promotes compassionate behaviour {{f}}. Affective empathy contributes to prosocial behaviour through actions of comforting others, acts of kindness and offering support (Isrealashvili, 2020). Affective empathy is distinguished from cognitive empathy because they involve two different psychological processes. A person may retain one form of empathy stronger than the other, meaning they might accurately understand what another person is feeling without without mirroring those emotions demonstrating cognitive empathy with a weaker display of affective empathy {{f}}. Alternatively, one might feel emotional discomfort in response to a friends' suffering without entirely understanding the reasons for it (Eisenberg, 2020). Affective empathy is often considered as a motivator of helpful behaviour as people who connect with someone else's distress are often motivated to ease that distress for the person and themselves {{f}}. However, excessive affective empathy can cause emotional exhaustion and empathetic distress. For this reason, it is argued that a balance between cognitive and affective empathy is crucial for effective and sustainable responsive behaviours (Cox et al., 2012). == Empathy and Prosociality == Prosociality is the tendency to perform behaviours that benefit the welfare of other people (Reniers et al., 2022). Such behaviours are voluntary and intended to help, support or cooperate with others. Prosociality develops {{f}} childhood and follows through to adolescence as individuals learn perspective taking skills, social norms and moral values {{f}}. Family relationships, education, social interactions and cultural influences all contribute to the development of prosocial behaviour {{f}}. Both cognitive and affective empathy are associated with prosocial behaviour {{f}}. Cognitive and affective empathy may contribute to prosociality through different pathways: understanding another person's needs versus emotionally responding to their experience. However, empathy does not automatically result in prosocial behaviour because emotion regulation, context, and the type of empathic response can influence whether someone helps {{f}}. It can be seen in Figure 3 {{ic|this should be Table 1) that affective empathy might result in the prosocial behaviour of either helping or withdrawal/escape. A study of social withdrawal implemented by Pool and Hassan (2026), concluded {{how}} that social withdrawal depended of {{g}} several difference factors. They explained that unsociable people were good at understanding others' perspectives and staying emotionally calm, displaying high levels of cognitive empathy {{huh}}. People that {{g}} were shy were still able to help those with an effected psychological state but could feel the emotional distress of others, presenting strong qualities of affective empathy. Those who were avoidant in their daily lives, generally care less for providing a positive prosocial contribution to those who are in emotional distress, displaying a withdrawal outcome as an effect of empathy. One construct of empathy alone cannot result in prosocial behaviour but when combining cognitive and affective empathy, there are higher results in positive social outcomes {{f}}. {| class="wikitable" |+Figure 3. Cognitive and affective empathy as distinct but interaction pathways to prosociality {{f}}. |'''Cognitive empathy''' |'''Affective empathy''' |- |Understanding another person's perspective |Emotional response to another person's state |- |Identifying their needs |Empathic concern or personal distress |- |Potentially appropriate helping |Helping or withdrawal/escape |} ==Learning features== '''Table 1''' {{ic|Caption Goes Here}} {| class="wikitable" style="margin: auto; |- !Dimension ! Cognitive empathy !! Affective empathy |- | Core process || Understanding another person's mental/emotional state. || Experiencing an emotional response to another person's state. |- |Related concept |Perspective-taking / mentalising. |Emotional sharing / experience-sharing. |- |Example |“I understand why they feel hurt.” |“I feel upset because they are hurting.” |- |Potential contribution to prosociality |Helps identify another person's perspective and needs. |Can generate emotional motivation to care and help. |- |Potential limitation |Understanding someone does not necessarily motivate helping. |Emotional sharing can produce personal distress rather than. |} Source: [https://link.springer.com/content/pdf/10.1007/s10803-026-07315-2.pdf Double Empathy Problem], [https://pmc.ncbi.nlm.nih.gov/articles/PMC3017348/ Empathy Related Responding] {{ic|Use the [[Help:Quiz]] tool}} You see a stranger in public crying. What would you think, feel and/or do? * Think = cognitive empathy * Feel = affective empathy * Do = Prosocial behaviour == Conclusion == Cognitive and affective empathy are two important but distinct components of empathy that contribute to prosocial behaviour. Cognitive empathy allows individuals to understand another persons' perspective, thoughts, and feelings, while affective empathy allows one to mirror those feelings themselves. Both can result is {{sp}} a person helping, comforting and supporting another. However, Empathy {{ic|remove unnecessary capitalisation; check throughout}} does not always lead to prosocial behaviour as personal distress and emotional overwhelm can cause withdrawal rather than help. Understanding these difference highlights that ultimately, both frames of empathy can play an important role in promoting prosocial behaviour when paired together. The most effective prosocial outcomes occur when cognitive and affective empathy work together. Understanding the the perspective of another while also experiencing appropriate emotional concern, enables individuals to provide support that is both compassionate and responsive to individualised needs. A balance between cognitive and affective empathy is essential in fostering positive social relationships and prosocial behaviour. ==See also== * [[wikipedia:Empathy|Empathy]] (Wikiversity) * [[Motivation and emotion/Lectures/Individual emotions|Individual emotions]] (Wikiversity) ==References == {{Hanging indent|1= Beadle, J., sheehan, A., Dahlben, B., & Gutchess, A. (2013). Aging, Empathy, and Prosociality. Oxford Academic, 70(2).https://academic.oup.com/psychsocgerontology/article-abstract/70/2/213/572414?redirectedFrom=PDF Cox, C. L., Uddin, L. Q., Di Martino, A., Castellanos, F. X., Milham, M. P., & Kelly, C. (2012). The balance between feeling and knowing: Affective and cognitive empathy are reflected in the brain’s intrinsic functional dynamics. Social Cognitive and Affective Neuroscience, 7(6), 727–737. https://doi.org/10.1093/scan/nsr051 Eisenberg, N., Eggum, N. D., & Di Giunta, L. (2010). Empathy-related Responding: Associations with Prosocial Behavior, Aggression, and Intergroup Relations. Social issues and policy review, 4(1), 143–180. https://doi.org/10.1111/j.1751-2409.2010.01020.x Fellowes, S. (2026). Cognitive Empathy, Affective Empathy and Empirical Research on the Double Empathy Problem: A Critical Methodology Review.https://link.springer.com/content/pdf/10.1007/s10803-026-07315-2.pdf Fink, E., & de Rosnay, M. (2023). Examining links between affective empathy, cognitive empathy, and peer relationships at the transition to school. ''Social Development'', ''32''(4), 1208–1226. <nowiki>https://doi.org/10.1111/sode.12685</nowiki> Fiveable. (2026, March 4). Empathy and prosocial behavior. Fiveable. https://fiveable.me/adolescent-development/unit-5/empathy-prosocial-behavior/study-guide/nX0hzv5WnNyAKG01 Greater Good Magazine. (2009). Empathy definition {{!}} what is empathy. Greater Good. https://greatergood.berkeley.edu/topic/empathy/definition Israelashvili, J., Sauter, D., & Fischer, A. (2020). Two facets of affective empathy: Concern and distress have opposite relationships to emotion recognition. Cognition and Emotion, 34(6), 1–11. https://doi.org/10.1080/02699931.2020.1724893 Khalid, A., Tarlinton, J., Kwan, T., & Khoo, K. (2023). Amiodarone-Induced Delirium. ''TTU Journal of Biomedical Sciences'', ''2''(1), 29–32. <nowiki>https://doi.org/10.53901/tjbs.2023.08.art03</nowiki> Maibom, H. (2020). Empathy (1st edn). Routledge. https://www-taylorfrancis-com.ezproxy.canberra.edu.au/reader/read-online/813e55c8-e789-40d2-9bea-a2da2105925b/book/epub?context=ubx Markus Paulus. (2015). ''Sugar and spice, and everything nice : exploring prosocial development through infancy and early childhood'' (C. Moore, M. Paulus, & A. Williams, Eds.). Frontiers Media SA. Merritt, Z. (2026, July 2). What is cognitive empathy and why it matters? Sicotests. https://www.sicotests.com/what-is-cognitive-empathy-and-why-it-matters/ Price, D. N. (2023). Author post: Cognitive empathy: What it is, why it matters. Forbes. https://www.forbes.com/sites/forbesbooksauthors/2023/08/14/cognitive-empathy-what-it-is-why-it-matters/ Reniers, R. L. E. P., Abu-Akel, A., & Seara-Cardoso, A. (2022). Editorial: Cognitive empathy and perspective taking: Understanding the mechanisms of normal and abnormal experiences and abilities. Frontiers in Psychiatry, 13. https://doi.org/10.3389/fpsyt.2022.945258 Sanders, J. J., Dubey, M., Hall, J. A., Catzen, H. Z., Blanch‐Hartigan, D., & Schwartz, R. (2021). What is empathy? Oncology patient perspectives on empathic clinician behaviors. Cancer, 127(22), 4258–4265. https://doi.org/10.1002/cncr.33834 Scott, R., Poole, K. L., Schmidt, L. A., & Hassan, R. (2025). Caring too much or too little? Relations among motivations for social withdrawal, empathy, and prosociality in emerging adulthood. Scandinavian Journal of Psychology, 66(5), 717–728. https://doi.org/10.1111/sjop.13116 }} ==External links== * [https://www.youtube.com/watch?v=3AgvKJK-nrk The Roots of Empathy and Compassion] (GGSC) * [https://www.youtube.com/watch?v=vhtACD8Nk7c Pondering Prosocial Behavior] (youtube.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Empathy]] rxsnge6mo1ja0aqa8fubsjbebh81v9i 0 331601 2844081 2843303 2026-11-03T02:49:05Z Jtneill 10242 2844081 wikitext text/x-wiki {{title|Affect heuristic:<br>What is the affect heuristic and how does it influence decision making?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} ;Scenario: A frightening headline and a familiar product Mia is shopping online for sunscreen she has used before. That morning, she sees a dramatic social media post claiming an ingredient in the product is dangerous. The post contains an alarming image but no risk estimate or link to the original evidence. Mia immediately feels uneasy and decides the sunscreen must be unsafe. Later, she reads a health agency explanation, checks the evidence and learns that the post left out important context. Her initial reaction was not foolish: it was a quick affective signal. The problem was treating that feeling as if it were a complete risk assessment. {{RoundBoxBottom}} The '''affect heuristic''' is the tendency to use a rapid positive or negative feeling associated with an object, activity, person or message as a shortcut when making judgements. A ''heuristic'' is a mental shortcut that reduces effort when information is incomplete or a decision is complex. ''Affect'' refers to the felt quality of “good” or “bad”; it can be subtle and quick rather than a strong emotion such as anger or fear (Slovic et al., 2007). The affect heuristic can influence judgements about risk, benefit, trustworthiness and choice before a person has deliberately analysed all available evidence. The practical problem is that a feeling may be informative, misleading, or both. Feelings can summarise experience efficiently, but they can also be shaped by vivid images, framing, mood, familiarity and irrelevant recent events. This chapter explains how the affect heuristic works, compares related theories, evaluates key research, and considers how people can combine emotional information with careful analysis. <div style="background-color:#e6f2ff; border:1px solid #99c9ff; border-radius:12px; padding:12px 18px; margin:10px 0;"> '''Focus questions''' * What is the affect heuristic, and how is it different from deliberate risk calculation? * How do feelings-as-information, risk-as-feelings and dual-process perspectives explain affective judgements? * What does research show about risk, benefit, time pressure and consumer choice—and what are its limitations? * When can affect help decision making, and when can it mislead? * What practical steps help people use feelings without mistaking them for evidence? </div> == How the affect heuristic works == [[File:Affect heuristic chapter figure.png|thumb|center|500px|'''Figure 1'''. Conceptual model of the affect heuristic. Emotional impressions influence perceived risks, benefits, and decisions. An affective impression can influence perceived benefits and risks, which in turn shape judgements and choices. The pathways are simplified: experience, context and analytical thought can also affect each stage.]] Affect is not identical to a fully developed emotion {{f}}. A person can experience a general positive or negative impression without being able to name a discrete emotion or explain its source {{f}}. A familiar brand may “feel right,” while a frightening photograph may make a technology feel dangerous. Slovic et al. (2007) describe affect as a rapid evaluative feeling that can become available for use in judgement. The heuristic occurs when this general feeling is used as information about the target. Rather than calculating every possible benefit, harm, likelihood and consequence, a person may ask—often without awareness—“How do I feel about this?” {{g}} The answer can influence several judgements at once. If a technology is viewed positively, its benefits may appear greater and its risks smaller; if it is viewed negatively, its risks may seem greater and its benefits smaller. This is important because objective risk and benefit are not necessarily opposites: a technology can have substantial benefits and substantial risks at the same time. {{f}} For example, someone who likes a new fitness tracker might assume it is accurate, useful and safe without checking independently. Someone who distrusts the company may infer the opposite. Neither impression proves the tracker is good or bad. The feeling may reflect genuine experience, but it may also come from advertising, a friend’s story or general mood. The heuristic describes a tendency, not a rule that every person follows in every situation. {{f}} The process is influenced by the decision-maker, task and setting. People differ in experience, knowledge, emotional responsiveness and willingness to rely on intuition. A familiar low-stakes choice may be handled efficiently through affect; a high-stakes unfamiliar decision deserves more scrutiny {{f}}. Time pressure and cognitive load can also make deliberation harder, increasing the influence of readily available impressions (Finucane et al., 2000; Shiv & Fedorikhin, 1999). == Theories that explain affective judgement == {{expand}} === The affect heuristic and the affect pool === Slovic and colleagues {{f}} propose that people draw on an accessible store of positive and negative associations when evaluating options. The “affect pool” is a metaphor for stored impressions built from personal experience, learned associations, imagery and social information (Slovic et al., 2007). When a decision is complex, consulting an overall feeling may be easier than retrieving and weighing many separate pieces of information {{f}}. Finucane et al. (2000) provided influential {{ic|type of?}} evidence by examining {{how}} judgements of risks and benefits associated with technologies. Their findings supported the idea that risk and benefit ratings are linked to overall affect, and that the inverse association became stronger under time pressure. This fits the theory: when there is less opportunity for deliberate analysis, a general impression can play a larger role. It does not mean affect is the only cause of risk judgements, or that every risk-benefit judgement is biased. Prior knowledge, values and actual characteristics also matter. === Feelings-as-information theory === Feelings-as-information theory proposes that people may use current feelings as information when forming judgements (Schwarz, 1990; Schwarz & Clore, 1983). A mood can be treated as a signal about the thing being judged, even when it arose from another source. If someone feels cheerful while evaluating a product, the product may receive a more positive assessment; if they feel anxious, it may be evaluated more negatively. People do not automatically treat every feeling as evidence about the target. When they recognise that their mood comes from an unrelated cause—such as a stressful commute—they may discount it. This is attribution: the person asks, explicitly or implicitly, what the feeling is about. Noticing a feeling and identifying its possible source can therefore reduce its inappropriate influence. This {{what}} theory complements the affect heuristic: the heuristic describes the shortcut, while feelings-as-information explains one way feelings acquire informational meaning. {{f}} === Risk-as-feelings and dual-process perspectives === Risk-as-feelings theory distinguishes immediate affective responses to danger from more deliberate assessments of probability and consequences (Loewenstein et al., 2001). A person may know that an event is unlikely yet feel intensely afraid of it; conversely, a familiar risk may feel harmless despite meaningful statistical danger. Such discrepancies matter because behaviour is influenced by how a risk feels, not only by calculated probability. {{f}} Slovic et al. (2004, 2005) contrast an experiential mode—fast, intuitive and associated with images and feelings—with an analytic mode involving rules, probabilities and conscious reasoning. These are useful descriptions of different processes, not necessarily two fully separate systems. Affective and analytic processes can interact: feelings direct attention, analysis revises initial impressions, and new information changes how an option feels. {{f}} Consumer research illustrates this interaction. Shiv and Fedorikhin (1999) found {{how}} that when cognitive resources were constrained, affectively attractive options had greater influence on choice; when resources were more available, deliberative considerations played a larger role. This suggests context can shift the relative influence of affect and cognition. It does not establish that intuitive choices are always poor or that deliberation always produces the best outcome. Expertise can make fast judgements effective, while lengthy analysis can be distorted by poor data or motivated reasoning. Emotion is not one uniform influence. Anger, fear, sadness and happiness can have different effects because they differ in perceived control, certainty and appraisals of responsibility (Lerner & Tiedens, 2006; Lerner et al., 2015). The affect heuristic focuses on the broad positive or negative feeling attached to a target. Discrete emotions may influence judgement through additional pathways. The theories overlap but answer different questions: the affect heuristic describes a shortcut; feelings-as-information explains how feelings are interpreted; risk-as-feelings emphasises immediate responses to risk; and dual-process perspectives describe the interaction of intuitive and analytic thought. {{f}} == What research tells us == {{expand}} === Risk and benefit judgements === Alhakami and Slovic (1994) reported that perceived risk and benefit were inversely related and that this relationship was associated with affective evaluations. Finucane et al. (2000) extended this work with experiments that altered information about the risks or benefits of technologies. Judgements of the unmanipulated dimension shifted too, consistent with the idea that an overall affective impression can influence both risk and benefit judgements. Their time-pressure findings suggested that the association became stronger when analytic deliberation was constrained. This evidence is valuable because experimental manipulations provide stronger evidence for causal influence than a simple correlation between mood and judgement. However, laboratory tasks and rating scales do not capture every feature of real decisions. Participants may interpret technologies differently, rely on prior beliefs, or respond to the experimental setting. A stronger relationship between risk and benefit ratings does not prove that every participant used an affective shortcut, nor that their final choices were necessarily wrong. The findings support affect as one influence among several. {{f}} === Risk perception beyond numbers === Research on risk perception shows that people respond not only to statistical probability but also to dread, familiarity, perceived control, catastrophic potential and the images associated with a hazard (Slovic, 1987; Slovic et al., 2004). A rare but vivid danger can feel more important than a common, less vivid one. These responses may be understandable: vivid examples can alert people to severe outcomes, and public concern can draw attention to risks that institutions have overlooked. But vividness is not a reliable measure of frequency. A dramatic story may dominate judgement even when it is not representative. Slovic et al. (2005) reviewed work {{ic|how many studies?}} on affect, risk and decision making and argued that feelings guide both risk perception and risk management. This review connects laboratory findings with real-world risk communication. However, a conceptual review does not provide one pooled estimate of the affect heuristic across all situations. Effects vary with the hazard, population, information format and decision being measured. The evidence is strongest for the general proposition that affect contributes to judgements, not for a claim that affect always overrides statistical information. === Consumer choice and limited resources === Shiv and Fedorikhin’s (1999){{ic|what sort of?}} experiments suggest cognitive load can change the balance between affective attraction and deliberation. In everyday settings, fatigue, distraction, multitasking or a crowded shop may make it harder to compare products carefully. Packaging, music, familiar branding and attractive presentation may then exert more influence. This does not mean an attractive product is chosen only because of emotion: price, convenience, quality and prior preferences may also shape choice. {{f}} A useful distinction is between a feeling that summarises relevant learning and one triggered by irrelevant cues. A positive reaction based on repeated reliable experience can be informative. A positive reaction created by polished advertising is not necessarily evidence of quality. The same applies to negative feelings: unease may alert someone to a real problem, but it can also reflect unfamiliarity or a misleading message. Better decisions require testing the feeling against the reasons and evidence behind it. {{f}} === Emotion is not always the enemy of reason === Lerner et al. (2015) reviewed research showing that emotions influence decision making through attention, memory, risk perception and preferences. The effects are not uniformly harmful. Emotion can prioritise important information, communicate values and support rapid responses. Fear may help a person attend to a credible warning; compassion may motivate help; regret can encourage learning. The issue is not whether emotion is present, but whether it is relevant and proportionate to the decision. {{f}} This broader literature cautions against treating the affect heuristic as a complete explanation for all emotional decision making. A general positive or negative impression is one pathway. Discrete emotions can have different effects, and culture, individual differences and decision structure can moderate outcomes. Critical evaluation therefore asks which affective response matters, for whom, in what situation, and with what measurable consequence. == When the affect heuristic helps—and when it misleads == Affect can help when choices are familiar, consequences are modest, and the feeling reflects useful experience {{f}}. A person may quickly recognise that a route feels unsafe because they have repeatedly noticed poor lighting or threatening behaviour. A clinician’s intuitive concern may prompt a more thorough assessment, provided it is checked rather than treated as a diagnosis. Affect can be a signal that directs attention to relevant information. Affect can mislead when driven by a vivid but unrepresentative story, unrelated mood, social pressure, advertising or emotionally loaded description {{f}}. Positive affect can make benefits seem larger and risks smaller; negative affect can do the reverse. This is consequential for health, money, safety and public policy, where probability and long-term outcomes matter. A feeling can be psychologically real without being an accurate estimate of external risk. {{f}} The same information can produce different feelings depending on presentation {{f}}. “Nine out of ten people benefit” may feel more reassuring than “one out of ten people do not benefit,” even when the numbers describe the same outcome. Framing effects are not identical to the affect heuristic, but they can interact by changing the impression a message creates {{f}}. A photograph of one accident can likewise make a rare event feel common. Decision quality depends on the relationship between feeling, evidence and context—not on a rule to always trust or distrust intuition {{f}}. == Applying the evidence to everyday decisions == The goal is not to eliminate emotion. It is to use emotion as a prompt for reflection while checking whether it is appropriate evidence for the decision. # '''Name the feeling.''' Ask: “What am I feeling about this option—interest, fear, excitement, disgust or reassurance?” Naming it creates a pause and makes the signal easier to examine. {{f}} # '''Check its source.''' Ask whether the feeling comes from direct experience, reliable evidence, a recent story, another person’s opinion, fatigue or general mood. If the source is unrelated, give the feeling less weight. {{f}} # '''Separate probability from vividness.''' Look for base rates, absolute risks, comparison groups and source quality. A dramatic example can matter without being typical. {{f}} # '''Compare benefits and harms independently.''' List benefits, risks, uncertainties and alternatives separately rather than assuming an option with high benefits must have low risks. {{f}} # '''Slow down for high-stakes choices.''' When practical, avoid major decisions under unnecessary time pressure, intense emotion or distraction. Seek independent information or a second opinion. {{f}} # '''Keep the feeling as a signal, not a verdict.''' If a strong reaction persists after checking facts, consider what relevant experience or value it might reflect. If it conflicts with reliable evidence, investigate the discrepancy. {{f}} Before sharing an alarming health post, a reader can check the original source, compare the claim with a credible health authority, and distinguish evidence of a hazard from the size of the actual risk. Before buying an expensive item, a consumer can notice excitement but compare specifications, independent reviews, price and return conditions. In each case, a feeling can start the enquiry; it should not end it. {{ic|Use a feature box}} {{ic|Use the [[Help:Quiz]] tool}} '''Scenario:''' Jordan sees a viral video showing a dramatic incident involving an electric scooter. Jordan immediately concludes that electric scooters are generally too dangerous to use. Which response best applies the chapter’s evidence? '''Question:''' Which response best applies the chapter’s evidence? * '''A.''' Keep the conclusion because a vivid example is more memorable than statistics. * '''B.''' Check reliable injury-rate data, consider exposure and severity, compare alternatives, and then decide how the evidence applies. * '''C.''' Ignore the emotional reaction completely because feelings never contain useful information. <div class="mw-collapsible mw-collapsed" style="border:1px solid #aaa; padding:0.7em; margin:0.5em 0;">'''Reveal the answer''' <div class="mw-collapsible-content">'''B is the best answer.''' It treats the reaction as a cue to investigate and checks rates, context and alternatives. A confuses vividness with frequency; C wrongly assumes affect is always useless.</div> </div> '''Reflection questions''' * Think of a recent decision where a strong feeling shaped your first judgement. What caused that feeling? * What evidence would help you decide whether the feeling was informative? * Would the decision be different if you were tired, rushed or exposed to a different headline? == Conclusion == The affect heuristic is a mental shortcut through which positive or negative feelings influence judgements of risk, benefit and choice. Research on risk-benefit judgements, time pressure and consumer decisions supports the view that affect can shape judgement, particularly when deliberation is constrained. Feelings-as-information and risk-as-feelings theories help explain how affect becomes influential, while dual-process perspectives highlight its interaction with deliberate analysis. The evidence does not support the simplistic conclusion that emotion is irrational or should always be ignored. Affect can efficiently summarise relevant experience and direct attention to important concerns. It can also be shaped by vividness, framing, mood and irrelevant associations. The practical lesson is to notice feelings, examine their source, and check important judgements against independent evidence. Used in this way, affect can contribute to better decisions without being mistaken for a complete assessment of risk. == See also == * [[w:Affect heuristic|Affect heuristic]] (Wikipedia) * [[w:Risk perception|Risk perception]] (Wikipedia) * [[w:Decision-making|Decision-making]] (Wikipedia) == References == {{Hanging indent|1= Alhakami, A. S., & Slovic, P. (1994). A psychological study of the inverse relationship between perceived risk and perceived benefit. ''Risk Analysis, 14''(6), 1085–1096. https://doi.org/10.1111/j.0272-4332.1994.tb00086.x Finucane, M. L., Alhakami, A., Slovic, P., & Johnson, S. M. (2000). The affect heuristic in judgments of risks and benefits. ''Journal of Behavioral Decision Making, 13''(1), 1–17. [https://doi.org/10.1002/%28SICI%291099-0771%28200001/03%2913%3A1%3C1%3A%3AAID-BDM333%3E3.0.CO%3B2-S DOI link] Lerner, J. S., & Tiedens, L. Z. (2006). Portrait of the angry decision maker: How appraisal tendencies shape anger’s influence on cognition. ''Journal of Behavioral Decision Making, 19''(2), 115–137. https://doi.org/10.1002/bdm.515 Lerner, J. S., Li, Y., Valdesolo, P., & Kassam, K. S. (2015). Emotion and decision making. ''Annual Review of Psychology, 66'', 799–823. https://doi.org/10.1146/annurev-psych-010213-115043 Loewenstein, G. F., Weber, E. U., Hsee, C. K., & Welch, N. (2001). Risk as feelings. ''Psychological Bulletin, 127''(2), 267–286. https://doi.org/10.1037/0033-2909.127.2.267 Schwarz, N. (1990). Feelings as information: Informational and motivational functions of affective states. In E. T. Higgins & R. F. Sorrentino (Eds.), ''Handbook of motivation and cognition: Foundations of social behavior'' (Vol. 2, pp. 527–561). Guilford Press. Schwarz, N., & Clore, G. L. (1983). Mood, misattribution, and judgments of well-being: Informative and directive functions of affective states. ''Journal of Personality and Social Psychology, 45''(3), 513–523. https://doi.org/10.1037/0022-3514.45.3.513 Shiv, B., & Fedorikhin, A. (1999). Heart and mind in conflict: The interplay of affect and cognition in consumer decision making. ''Journal of Consumer Research, 26''(3), 278–292. https://doi.org/10.1086/209563 Slovic, P. (1987). Perception of risk. ''Science, 236''(4799), 280–285. https://doi.org/10.1126/science.3563507 Slovic, P., Finucane, M. L., Peters, E., & MacGregor, D. G. (2004). Risk as analysis and risk as feelings: Some thoughts about affect, reason, risk, and rationality. ''Risk Analysis, 24''(2), 311–322. https://doi.org/10.1111/j.0272-4332.2004.00433.x Slovic, P., Peters, E., Finucane, M. L., & MacGregor, D. G. (2005). Affect, risk, and decision making. ''Health Psychology, 24''(4S), S35–S40. https://doi.org/10.1037/0278-6133.24.4.S35 Slovic, P., Finucane, M. L., Peters, E., & MacGregor, D. G. (2007). The affect heuristic. ''European Journal of Operational Research, 177''(3), 1333–1352. https://doi.org/10.1016/j.ejor.2005.04.006 }} == External links == * [https://pubmed.ncbi.nlm.nih.gov/16045417/ Slovic et al. (2005), Affect, risk, and decision making (PubMed)] * [https://doi.org/10.1111/j.0272-4332.2004.00433.x Slovic et al. (2004), Risk as analysis and risk as feelings] * [https://www.psychologicalscience.org/journals/current-directions/j.1467-8721.2006.00461.x Slovic and Peters (2006), Risk perception and affect] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Affect]] [[Category:Motivation and emotion/Book/Cognition]] l4109vy51vggim5kju9xvljiksh83y9 1 331738 2844024 2831349 2026-11-02T21:16:51Z Jtneill 10242 Book chapter feedback 2844024 wikitext text/x-wiki ==Focus questions== Great progress. The titles and layout look very clear and well considered. The second and third questions don't have capitals at the beginning. --[[User:U3228742|U3228742]] ([[User talk:U3228742|discuss]] • [[Special:Contributions/U3228742|contribs]]) 11:08, 31 August 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:U3280159|Hi [[User:U3280159|U3280159]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:24, 4 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) <!-- Other ---> # The Overview and Conclusion should not use sub-headings (fixed) <!-- Alignment with focus questions --> # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Basic # Insufficient # Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # A scenario is planned; make sure it clearly illustrates the key difference # Put the scenario in a focus box (fixed) # Add an image to the scenario to help attract reader interest <!-- Description --> # A basic description of the problem/topic is planned or presented <!-- Focus questions --> # Promising focus questions # Develop closer alignment between the sub-title, focus questions, and top-level headings # Put the focus question in a feature box (fixed) |4= <!-- Key points--> <!-- Overall --> # Basic development <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Insufficient use of APA style citations <!-- GenAI ---> # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # A figure is planned, but poorly presented; see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Is this figure from [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Promising use of table(s) # Is this table from [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. # Include citation(s) for sources of information in Table 1 if human-drawn from other sources |7= <!-- References --> <!-- Overall --> # Basic # These are not the best 6 sources about this topic; a more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## author initials ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) |8= <!-- Resources --> <!-- See also --> # See also ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- External links --> # External links ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |9= <!-- User page --> # Excellent # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] <!-- Social contribution --> # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:24, 4 September 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an insufficient chapter # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic #* quality of written expression #* integrate more strongly with other chapters by using embedded links to related topics #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]] <!-- Overall – Citations --> # Insufficient use of academic, peer-reviewed citations to support claims <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FCognitive_versus_affective_empathy&diff=2844023&oldid=2840877 these copy edits] |2= <!-- Overview – Comments... --> # Basic <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Basic scenario or case study # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) <!-- Overview – Scenario - Figure --> # The Figure 1 caption should explain more clearly how it is related to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Explains the psychological problem or phenomenon reasonably in a basic way <!-- Overview – Focus questions --> # The focus questions are clear and relevant)\ # See copy edits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement <!-- Theory – Builds on --> # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Basic use of tables, figures, and/or lists to clearly convey key theoretical information # Tables and figures should be embedded and integrated rather than presented at the end <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. <!-- Theory – Examples --> # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Basic summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised <!-- Conclusion – Focus questions --> # Address the focus questions <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is below professional standard. For example, many sentences do not start with a capital letter. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. <!-- Style – Sentences --> ## Consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression <!-- Style – Layout --> # Layout ## The chapter structure is underdeveloped; expand by using subheadings ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Integrate learning features rather than having it as a stand-alone section <!-- Style – Grammar --> # Grammar ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') <!-- Style – Copy editing --> # Copy editing ## This appears to be an incomplete draft; further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] <!-- Style – Figures --> ## Figures ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing without bold or italics. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Style – Tables --> ## Table(s) ### Insufficiently captioned ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Style – Tables - Citations --> ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use problematic [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## Do not include author first name or initials ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois <!-- Style – References --> ## References use reasonably good APA style: ### Only include references for sources that you consulted and read ### Check and correct italicsation ### Include dois where available ### Hyperlink dois for 1-click access ### More copy editing is needed to fix formatting inconsistencies |8= <!-- Learning features comments... --> # Basic use of learning features <!-- Learning features – Wikipedia embedded links --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Basic use of figure(s) # Reasonably good use of table(s) # No use of feature boxes other than the opening scenario # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # Make the quiz questions more interactive by using the [[Help:Quiz|quiz]] feature <!-- Learning features – See also --> # Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Also include links to related Wikipedia articles ## Add more links <!-- Learning features – External links --> # Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Link to the top 3-6 external resources about this topic |9= <!-- Social contribution comments... --> # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 21:16, 2 November 2026 (UTC) mauftzz0y7ix0cfe6eddmuoffjo028n 1 331777 2844041 2833128 2026-11-02T22:15:03Z Jtneill 10242 Book chapter feedback 2844041 wikitext text/x-wiki == Food for thought: Integrating Pavlov's classical conditioning == Hey there, Great job on developing your topic, I enjoyed reading your book chapter so far! I wondered whether you had considered incorporating concepts from Pavlov's [[wikipedia:Classical_conditioning|classical conditioning]], even briefly, to help contextualise fear-conditioning. From my understanding, fear-conditioning is a specific form of classical conditioning, so including Pavlov's theory may be relevant to your chapter. Best regards. [[User:U3285438|U3285438]] ([[User talk:U3285438|discuss]] • [[Special:Contributions/U3285438|contribs]]) 05:21, 2 September 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:ChillPsychGuy0607|Hi [[User:ChillPsychGuy0607|ChillPsychGuy0607]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:28, 8 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement # The structure/scope may be overly ambitious within the maximum word count; if so, be selective about what's essential to address the question <!-- Alignment with focus questions --> # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement (e.g., there are 5 focus questions) # There are probably too many focus questions (e.g., #4 isn't needed to address the question; this could be embedded elsewhere where relevant) <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Basic # Insufficient # Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # Promising scenario; I've added a sentence to the end # Avoid animated image in the scenario box (too distracting); use a static image instead <!-- Description --> # A clear description of the problem/topic is planned or presented; keep it simple; don't get overly detailed here <!-- Focus questions --> # Reasonably good alignment between focus questions and heading structure, but consider closer alignment |4= <!-- Key points--> <!-- Overall --> # Promising development # Focus on providing an integrative review of the most relevant theories and research on the topic <!-- Scope --> # The scope is about right, but it may be that all planned aspects cannot be reasonably covered within the book chapter word count; in that case, be selective and concentrate on key aspects that address the question in the sub-title <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Remove in-text bold # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # Consider using Studiosity Writing Feedback+ or a similar writing-support service (e.g., Grammarly) to improve the quality of written expression and check for grammatical and spelling errors in the book chapter draft. <!-- GenAI ---> # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Relevant figure(s) are presented and captioned <!-- Caption --> # Figure caption(s) could better explain how the image connects to key points being made in the main text <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider decreasing image size(s) to make them less dominant |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed linksto [[Motivation and emotion/Book|book chapters]]. <!-- Scenarios/examples/case studies --> # Focus scenarios on real-world situations (rather than summaries of research articles) <!-- Quiz --> # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages <!-- Tables --> # Promising use of table(s) # Include citations for source(s) of information in Table 1 |7= <!-- References --> <!-- Overall --> # Very good <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Citations --> # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## [[Help:Wikitext quick reference|italicisation]] |8= <!-- Resources --> <!-- See also --> # See also ## Good ## Use internal links (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Some links are incomplete (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- External links --> # External links ## Good ## Most of these links should be moved into References and cited ## Link here to the most relevant external non-peer-reviewed resources about this topic |9= <!-- User page --> # Very good <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # Use an internal link to the book chapter (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |10= <!-- Social contribution --> # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:28, 8 September 2026 (UTC) == Prediction Error and Extinction == Hi, Great start to your chapter. I was attracted to the chapter given the title and subject matter. Once I got into it, I was also drawn on how close some of the theory aspects involved with your specific topic and mine were - I'm doing immersive therapy for PTSD treatment ([[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment]]) and spend some time looking at some of the psychological mechanisms, including prediction error. If you need additional references, I have an extensive list of relevant ones in my chapter references that may help you narrow the field. Good luck. [[User:StretchBeyond|StretchBeyond]] ([[User talk:StretchBeyond|discuss]] • [[Special:Contributions/StretchBeyond|contribs]]) 08:31, 14 September 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic chapter # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic #* quality of written expression #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words <!-- Overall - Edit summaries --> #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits <!-- Overall - GenAI --> #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|Motivation and emotion/Wikiversity/Edit summary]], with a link to the chatbot conversation, when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]] <!-- Overall – Citations --> #Basic use of academic, peer-reviewed citations to support claims <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FFear_extinction&diff=2844040&oldid=2841178 these copy edits] |2= <!-- Overview – Comments... --> # Very good <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Basic scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 lacks sufficient relevance to the scenario # Figure 1 was too large (fixed) <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Good—key theories are quite explained and applied; minor to moderate areas for improvements <!-- Theory – Builds on --> # Builds in a basic way on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Good use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Basic integration between the most relevant theory(ies) and the best research # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Good summary and conclusion # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is basic, with several aspects below professional standard mainly because the chapter appears to be based on overuse of insufficiently acknowledged genAI content; a more convincing chapter would either use more human-written words and/or use finer-grained acknowledgement of genAI with publicly available links to the chatbot conversations as per the [[Motivation and emotion/Assessment/Using generative AI|unit's using genAI guidelines]]. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ## There is considerable "wheel spinning" (repetive content) which is common with overuse of genAI <!-- Style – Paragraphs --> ## Bullet points are overused. Use full sentences and paragraphs. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid directional phrases such as “as previously mentioned” or “as noted above”, which add little value and can interrupt the flow of writing. Where cross-referencing is necessary, use [[w:Help#Section linking|section links]] to guide readers to the relevant content. <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections <!-- Style – Grammar --> # Grammar ## The grammar for many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – Figures --> ## Figures ### Well captioned <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). <!-- Style – Tables --> ## Table(s) ### Well ### Use APA style for the caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) ### The source(s) of the information presented in one or more tables needs acknowledgement <!-- Style – Tables - Citations --> ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## Do not include author first name or initials ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois <!-- Style – References --> ## References use very good APA style: ### Check and correct use of italicisation <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Reasonably good use of learning features <!-- Learning features – Wikipedia embedded links --> # Basic use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Basic use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Reasonably good use of figure(s) # Reasonably good use of table(s) # Basic use of feature box(es) # Reasonably good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Reasonably good use of quiz(zes) and/or reflection question(s) # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end <!-- Learning features – See also --> # Insufficient use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] <!-- Learning features – External links --> # Basic use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite |9= <!-- Social contribution comments... --> # ~2 logged, useful, mostly moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~1 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:15, 2 November 2026 (UTC) frrygy9tftgwzhm9gl5s49fp7lleh23 1 331792 2844079 2842482 2026-11-03T02:40:55Z Jtneill 10242 2844079 wikitext text/x-wiki == Interest In subject == I found this topic really interesting and I think the chapter is developing really well. I also noticed a slight structuring error in your reference list so I made a small edit to ensure it remained consistent and improved the layout. Good luck with the rest of your chapter !!! [[User:Ckopplemann|Ckopplemann]] ([[User talk:Ckopplemann|discuss]] • [[Special:Contributions/Ckopplemann|contribs]]) 04:34, 3 September 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:LMM26|Hi [[User:LMM26|LMM26]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:07, 7 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – would benefit from further development (refinement) # The headings appear to focus on GM itself, rather than how to cultivate it <!-- Alignment with focus questions --> # Develop closer alignment between sub-title, focus questions, and top-level headings |3= <!-- Overview--> # Insufficient <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section # However, the scenario doesn't illustrate the development of a GM <!-- Description --> # The description is general about GM; it doesn't sufficiently focus on the ''development of a GM'' <!-- Style --> <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings # More specifically, revise the FQs to focus on the ''development of a GM'' |4= <!-- Key points--> <!-- Overall --> # Insufficient development due to lack of emphasis on theory, research, and examples of ''developing a GM'' <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings <!-- Conclusion --> # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # A figure is presented and captioned # The relevance of the figure to the topic is unclear <!-- Caption --> # Figure caption(s) could better explain how the image connects to key points being made in the main text <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Basic use of scenario/example/case study, but the scenario does not illustrate the topic <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Very good <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Very good ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Include source in brackets after link ## Use alphabetical order |9= <!-- User page --> # Used effectively <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:07, 7 September 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic to reasonably good chapter that makes good use of psychological theory and reasonably good use of research to address a real-world phenomenon or problem # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic #* quality of written expression <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits <!-- Overall – Citations --> # Good use of academic, peer-reviewed citations to support claims # The Reeve textbook is overused as a citation; go to primary sources <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FDeveloping_a_growth_mindset&diff=2842481&oldid=2839328 these copyedits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Very good—key theories are well explained and applied; minor areas for improvements <!-- Theory – Builds on --> # Builds effectively on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Very good use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Reasonably good review of relevant research # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # More detail about key studies would be ideal # Claims are reasonably well referenced, however use of evidence was quite superficial. Possibly only the abstracts of many cited sources were consulted (e.g., the neuroscience section focused on plasticity and relied on Ng (2018), but this paper barely mentions synaptic plasticity). <!-- Research – Critical thinking --> # Basic [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Reasonably good integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Basic summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # Address the focus questions <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is OK but there are several aspects which are below professional standard <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. <!-- Written expression – Layout --> # Layout ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] <!-- Written expression – Grammar --> # Grammar ## The grammar for many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Proofreading --> # Proofreading ## More proofreading is needed (e.g., fix punctuation and typographical errors) to bring the quality of written expression closer to a professional standard <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) <!-- Written expression – Figures --> ## Figures ### Very well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Tables --> ## Tables ### Very well captioned ### Table captions should use APA style (capitalisation) ### Each Table is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use very good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) <!-- Written expression – References --> ## References use reasonably good APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Very good use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Very good use of figure(s) # Good use of table(s) # Basic use of feature box(es) # Basic use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section <!-- Learning features – External links --> # Good use of the [[{{PAGENAME}}#External links|External links]] section ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] |9= <!-- Social contribution comments... --> # ~10 logged, useful, mostly minor to moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:17, 25 October 2026 (UTC) 9htnrmpbqb24wsmy30zp2k755a2iloh 1 331836 2844016 2835089 2026-11-02T20:25:22Z Jtneill 10242 Book chapter feedback 2844016 wikitext text/x-wiki == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:ChelsSchofield|Hi [[User:ChelsSchofield|ChelsSchofield]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:24, 5 September 2026 (UTC) |} :Thank you for the detailed marking. I will actively work towards these improvements. [[User:ChelsSchofield|ChelsSchofield]] ([[User talk:ChelsSchofield|discuss]] • [[Special:Contributions/ChelsSchofield|contribs]]) 00:58, 9 September 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement <!-- Alignment with focus questions --> # Develop closer alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Good <!-- Scenario --> # A scenario or case study is presented in a feature box at the start of this section # Add an image (not a diagram) to the scenario to help attract reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings |4= <!-- Key points--> <!-- Overall --> # Solid development # Include an introductory paragraph before branching into sub-sections <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Conclusion --> # Conclusion is well underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. ... (fixed) <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) # Place each quiz question in the most relevant section <!-- Tables --> # Promising use of table(s) |7= <!-- References --> <!-- Overall --> # Basic <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## [[Help:Wikitext quick reference|italicisation]] ## include hyperlinked dois |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Include source in brackets after link ## Use alphabetical order ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Very good <!-- Description about self --> # Excellent description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |10= <!-- Social contribution --> # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:24, 5 September 2026 (UTC) ==Comment== Hi! I really like how you've connected pain perception with different technologies and psychological theories. I noticed that parts of the “Understanding pain perception and management” section still read like planning notes, for example, “Relate this distinction to technology” and “Refer to increased attention towards bodily sensations.” It might help to turn these into complete paragraphs explaining the ideas directly to the reader. I think this would make the section flow more smoothly and clarify the connection between the psychological theory and technology. Hope this helps! [[User:U3239236|U3239236]] ([[User talk:U3239236|discuss]] • [[Special:Contributions/U3239236|contribs]]) 07:47, 29 September 2026 (UTC) :thank you [[User:ChelsSchofield|ChelsSchofield]] ([[User talk:ChelsSchofield|discuss]] • [[Special:Contributions/ChelsSchofield|contribs]]) 01:38, 30 September 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic chapter # The chapter seems to be unfinished (e.g., contains bullet points, a lot of typographical and grammatical errors, and is well over the maximum word count) <!-- Overall - Edit summaries --> #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits <!-- Overall - GenAI --> #* The links to genAI conversations used to generate content are not provided in [[Motivation and emotion/Wikiversity/Edit summaries|Motivation and emotion/Wikiversity/Edit summaries]] as required by the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] <!-- Overall – Word count --> # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FTechnology-based_pain_management&diff=2844015&oldid=2839404 these copy edits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are good # The focus questions could be improved by: ## matching the top-level headings more closely |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Good # Be more selective about which technologies and theories to concentrate on (pick the main ones), to provide a more cogent chapter within the word count limit; too much is not better; practice the skill of selectivity # Some of the theory is not pain-related <!-- Theory – Builds on --> # Builds on two [[w:|Wikipedia]] article but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Very good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Basic use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In some places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. <!-- Theory – Examples --> # Very good use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Basic review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # More recent research would be ideal # In some places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. <!-- Research – Critical thinking --> # Basic [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Basic integration between the most relevant theory(ies) and the best research # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI may be over-used; write more in your own words, based on your own knowledge based on reading the primary sources <!-- Style – Paragraphs --> ## Bullet points are overused. Use full sentences and paragraphs. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. <!-- Style – Layout --> # Layout ## The headings could be more clearly aligned with the focus questions ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Remove abbreviations/citations from headings ## Integrate learning features rather than having it as a stand-alone section <!-- Style – Grammar --> # Grammar ## The grammar for many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – Abbreviations --> ## Abbreviations ### Explain abbreviations on first use: Spell out each abbreviation when it first appears (e.g., self-determination theory (SDT)), to ensure readers understand the term before its subsequent use <!-- Style – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Style – Figures --> ## Figures ### Reasonably well captioned <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Style – Citations --> # Citations use poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)]; the APA style for citations is below professional standard # In some places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Parentheses are often incorrectly/unnecessarily used for citations # Spaces are often missing within citations <!-- Style – References --> ## References use good APA style: ### Remove numbers ### Check and correct use of italicisation ### References are not counted for marking purposes (over maximum word count) |8= <!-- Learning features comments... --> # Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Add more embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Basic use of figure(s) # No use of table(s) within maximum word count # No use of feature boxes other than the opening scenario # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # No use of quiz(zes) and/or reflection question(s) within the maximum word count <!-- Learning features – See also --> # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~3 logged, useful, mostly moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 20:25, 2 November 2026 (UTC) kr96qhnlu2c0tos1rucea9dxe1ugpnd 1 331989 2844074 2832790 2026-11-03T00:56:50Z Jtneill 10242 Book chapter feedback 2844074 wikitext text/x-wiki <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Heading structure --> <!-- 2-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement <!-- Conceptual --> # The sub-title suggests three main questions; consider aligning the headings more strongly with these questions # Consider renaming "Understanding emotional regulation" to "What is emotional dysregulation?" # Consider renaming "Assocation between ..." because this doesn't directly map to the sub-title questions # Be careful not to overly focus on: (a) disorders; (b) clinical treatments because neither of these are called for in the title or sub-title; psychopathologies and psychotherapies may be relevant but they are not sufficient to address the psychological topic, which is broader <!-- Other ---> # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- Alignment with focus questions --> # Develop closer alignment between sub-title, focus questions, and top-level headings |3= <!-- Overview--> # Very good <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section # Increase the image size <!-- Description --> # A clear description of the problem/topic is planned or presented # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. <!-- Style --> # Use present, rather than future, tense <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings |4= <!-- Key points--> <!-- Overall --> # Key points are well developed for each section # Excellent theoretical models; be selective (e.g., concentrate on top 2 or 3): * See also: https://www.tandfonline.com/doi/full/10.1080/1047840X.2014.940781#d1e276 <!-- Scope --> # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title <!-- Theory and research --> # Promising balance of theory and research <!-- Conclusion --> # Conclusion is well underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] <!-- Scenarios/examples/case studies --> # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages <!-- Tables --> # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Excellent ## Use [[w:Letter case#Sentence casing|sentence casing]] |9= <!-- User page --> # Excellent <!-- Description about self --> # Excellent description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:09, 11 September 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem <!-- Overall - Edit summaries --> #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits, but would ideally be more descriptive and extensive <!-- Overall – Citations --> # Excellent use of academic, peer-reviewed citations to support claims <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmotion_dysregulation&diff=2844073&oldid=2839578 these copy edits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Use an example that a wider audience can relate to; less than 5% of the world population are university students <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely (partially fixed) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used <!-- Theory – Examples --> # Excellent use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses ## To improve, report the number of studies; there's a big difference in the reliability of reviews involving a handful of studies vs. hundreds of studies # More detail about key studies would be ideal ## To improve, also report the size of relationships, not just the significance and direction # Claims are well referenced <!-- Research – Critical thinking --> # Excellent [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent integration between the most relevant theory(ies) and the best research |6= <!-- Conclusion comments... --> # Excellent summary and conclusion # Highlights the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Clear take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is excellent <!-- Style – Paragraphs --> ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. <!-- Style – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. <!-- Style – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections <!-- Style – Grammar --> # Grammar ## The grammar for some sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback <!-- Style – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] (also known as the Oxford comma) <!-- Style – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) <!-- Style – Figures --> ## Figures ### Very well captioned <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Style – Tables --> ## Table(s) ### Very well captioned <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Style – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) <!-- Style – References --> ## References use excellent APA style: |8= <!-- Learning features comments... --> # Excellent use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # No use of feature boxes other than the opening scenario # Reasonably good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Basic use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent use of the [[{{PAGENAME}}#See also|See also]] section <!-- Learning features – External links --> # Excellent use of the [[{{PAGENAME}}#External links|External links]] section |9= <!-- Social contribution comments... --> # ~36 logged, useful, mostly moderate, very well logged and summarised contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # Thanks very much for your extensive contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 3 November 2026 (UTC) gc0vpi4j6aubjh4di2oitenfjdnde4s 1 332032 2844153 2842779 2026-11-03T09:23:06Z Jtneill 10242 Remove 6 mentions of possible genAI use 2844153 wikitext text/x-wiki <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> <!-- 1-level --> # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Remove duplicate headings <!-- Alignment with focus questions --> # Develop closer alignment between sub-title, focus questions, and top-level headings |3= <!-- Overview--> # Basic <!-- Scenario --> # Promising scenario; could benefit from further development e.g., the scenario is somewhat abstract; how can it illustrate the core problematic more clearly? # Consider revising the scenario to make it more relatable to a wider international audience; university/college students represent only around 3% of the world's population # Add an image to the scenario to help attract reader interest <!-- Description --> # A basic description of the problem/topic is planned or presented # Introduce topic using plain English; most citations can be moved into subsequent setions # What are the most relevant psychological theories? <!-- Focus questions --> <!-- Alignment and quality --> # Develop closer alignment between the sub-title, focus questions, and top-level headings <!-- Key points--> <!-- Overall --> # Insufficient development # Provide more detailed edit summaries <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Insufficient use of citations <!-- Conclusion --> # Conclusion is underdeveloped # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # A figure is presented and captioned # The relevance of the figure to the topic is unclear <!-- Caption --> # Figure caption(s) could better explain how the image connects to key points being made in the main text <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Promising use of quiz question(s) # Place each quiz question in the most relevant section # Focus the quiz question(s) on the take-home messages <!-- Tables --> # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Basic <!-- Systematic reviews --> # At least one relevant systematic review and/or meta-analysis has been identified <!-- Citations --> # All references need in-text citation # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## [[Help:Wikitext quick reference|italicisation]] |8= <!-- Resources --> <!-- See also --> # See also # Links lack sufficient relevance to topic; select the most relevant links <!-- External links --> # External links ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Basic <!-- Description about self --> # Description about self provided # Note that you are able to negotiate an alternative assessment format if you do not wish to use Wikiversity; the earlier this is done, the better <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:08, 14 September 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words <!-- Overall - Edit summaries --> #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall – Citations --> # Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEffort_regulation_and_cost-benefit_decision-making&diff=2842778&oldid=2841190 these copyedits] |2= <!-- Overview – Comments... --> # Basic <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Reasonably engaging scenario or case study # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) <!-- Overview – Scenario - Figure --> # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Explains the psychological problem or phenomenon in a basic/general way # Convert bullet-points into sentences and paragraphs <!-- Overview – Focus questions --> # The focus questions are reasonably good # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## avoid overlapping ## splitting double-barrelled questions into separate questions |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Basic use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Basic review of relevant research # Basic use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links |6= <!-- Conclusion comments... --> # Reasonably good summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is basic <!-- Written expression – Layout --> # Layout ## The chapter structure is underdeveloped; expand by using subheadings ## Integrate review of research throughout, rather than relying on a dedicated section <!-- Written expression – Grammar --> # Grammar ## The grammar for many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] <!-- Written expression – Figures --> ## Figures ### Well captioned <!-- Written expression – Figures - Citations --> ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – Citations --> # In many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Written expression – References --> ## References use very good/ APA style: ### Check and correct use of italicisation |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Insufficient use of figure(s) # Basic use of table(s) # No use of feature boxes other than the opening scenario # Basic use of scenarios, case studies, or examples # As discussed in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]], images should be uploaded to [[c:Wikimedia Commons|Wikimedia Commons]] <!-- Learning features – Quizzes --> # Reasonably good use of quiz(zes) and/or reflection question(s) # Many quiz questions have grammatical errors <!-- Learning features – See also --> # Basic use of the [[{{PAGENAME}}#See also|See also]] section ## I remove several generic links ## Use alphabetical order <!-- Learning features – External links --> # Insufficient [[{{PAGENAME}}#External links|External links]] section ## Link to the top 3-6 external resources about this topic |9= <!-- Social contribution comments... --> # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:32, 27 October 2026 (UTC) 9kjhid886b9cn91xz650ak3lwu6l6e4 1 332064 2844185 2842769 2026-11-03T11:49:06Z Jtneill 10242 2844185 wikitext text/x-wiki <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Heading structure --> <!-- 2-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) # There may be conceptual overlap between some sections <!-- Alignment with focus questions --> # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment |3= <!-- Overview--> # Good <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section # I moved an image into the feature box to help attract reader interest <!-- Description --> # A promising description of the problem/topic is planned or presented # What are the most relevant psychological theories? <!-- Focus questions --> <!-- Alignment and quality --> # Reasonably good alignment between focus questions and top-level headings, but consider closer alignment # Avoid double-barrelled questions # There may be conceptual overlap between some questions; consider consolidating |4= <!-- Key points--> <!-- Overall --> # Promising development # Focus on providing an integrative review of the most relevant theories and research on the topic # Provide more detailed edit summaries <!-- Citations --> # Basic use of citations <!-- GenAI ---> # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Relevant figure(s) are presented and captioned <!-- Caption --> # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[Motivation and emotion/Book|book chapters]]. Also embed links to [[w:|Wikipedia]]. <!-- Scenarios/examples/case studies --> # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## [[Help:Wikitext quick reference|italicisation]] |8= <!-- Resources --> <!-- See also --> # See also ## Very good ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use alphabetical order <!-- External links --> # External links ## Very good ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Use alphabetical order <!-- User page --> # Used effectively <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:16, 15 September 2026 (UTC) == Suggestion about chapter == Hi, You made really good progress for your book chapter. I found Wallas’ Four-Stage Model of the Creative Process theory related to creativity, which I thought you can include in your book chapter. Please find reference below for the theory: Sadler-Smith, E. (2015). Wallas’ Four-Stage Model of the Creative Process: More Than Meets the Eye? Creativity Research Journal, 27(4), 342–352. https://doi.org/10.1080/10400419.2015.1087277 All the best. --[[User:P U3270518|P U3270518]] ([[User talk:P U3270518|discuss]] • [[Special:Contributions/P U3270518|contribs]]) 09:38, 16 September 2026 (UTC) == Reccomendations: Overview and headings == Hello, I really like this topic and I think you've encapculated the core components of it very well in your book chapter so far. As I am a super creative person, I love this topic. I have two main points of feedback, feel free to take pieces or ignore entirely. I am obviously a student (same as you) and am writing this feedback based on my own understanding of what the assessment requires. Hope this helps. :) For the overview, you brought up two distinct components of your topic (creativity and then effort) which is good but you may have gone on a little bit too long. I believe if you were to shorten your overview to just the scenario, you could get your point accross more succinctly. See my example of your scenario trimmed: "''You have been staring at a blank page for hours, trying to come up with an original idea for a creative project. Just when you decide to take a break, an idea suddenly comes to you. You feel excited and inspired, and you immediately start working. At first, everything seems to flow naturally and you feel confident about what you are creating. As you continue, however, the excitement begins to fade. The idea is harder to develop than you expected, and finishing the project requires concentration, persistence, and several changes along the way.'' Where did that initial creative spark go and how and why does one persevere despite it?" Then I would proceed to add your explanation beneath the scenario. I did a little rewording to help clarify your key points. "''This experience shows how inspiration can provide a starting point for creativity but also that creativity does not depend on inspiration alone, particularly in its execution. Inspiration can help people recognise new possibilities and motivate them to turn those possibilities into action (Oleynick et al., 2014) while developing the initial idea can require continual effort, revision, problem-solving, and persistence. This raises an interesting question: Does inspiration drive us to put effort into creative work, or can the effort we put into creating also lead to new inspiration?"'' I did remove one of your sources as I found it detracted from the actual concepts you were discussing in your scenario. I'm sure it'd be very easy to reuse it in the body of your chapter so don't stress. My next major piece of advice actual involves your heading and chapter structure. This is not to say I believe all the work you have done is void, but I believe reorganising and clarifying creative features will really strengthen your chapter (aforementioned, I'm very passionate about this topic lol) Here's my suggestion on a potential structure shift. (H = heading, SH = subheading) H1 Creativity SH1 Creativity and humanity. Here is where I would outline what creativity actually is. What mechanisms define it? Why do we do it? (socially, culturally, etc) This can be really broad and involve pretty much anything you want but I believe starting with a strong explanation of this key component will really strengthen your chapter overall. Again, it may seem obvious, but the concept itself is very difficult to pin down in one classification. SH2 Creativity and inspiration. This is where I would discuss what it means to be creatively inspired. I would link this back to your original paragraph in your first heading+ this point you made after your scenario. "'''''Creative inspiration can give people new ideas and motivate them to start creating.''' It can also help people see new possibilities and feel more interested in developing an idea. Inspiration can connect the discovery of an idea with the motivation to pursue it. This is reflected in the distinction between being “inspired by” something and being “inspired to” act on it (Oleynick et al., 2014).''" You may use that piece as this headings finisher perhaps. H2 Effort SH1 Effort. What is it and why is it important? Again, rinse and repeat above. What is effort? Why is it hard? Why do we feel inclined to do it more in certain situations than others? I would recoommend linking in motiviational theories (Perhaps something to do with goals?) for effort or tie it into other models discussed in class when your explain it. I would try to focus more on the motivational explanations here rather than direct application to creativity, for fear of sounding repetitive throughout the chapter. SH2 How effort applies to creativity. Applying the above explanation to creativity. I would definitly tie in some motiviational theories here, particularly to do with success and goals. H3 Insipration - the middle ground between creativity and effort (or some such title, you write what you like). You probably don't need subheadings for this (really depends on your word count so far I guess) but this is where I would discuss the interchangeable socioeconomics of inspiration on creativity/effort/persistence (which is a component you've mentioned a few times in your book chapter so far. When I say socioeconomics, I don't mean the impact of one on the other (I would save that for my final heading/conclusion) but rather the manifestation of them. A theoretical case study could be a good way to do this. Say there are three people, one with objectively high amounts of inspiration, one with objectively higher amounts of creativity, and one with objectively higher amounts of effort. Posing questions to your readers such as "in a task of putting together a lego set, which person do you think would complete the task first?" Would give them a chance to apply the knowledge they've learnt recently. You can then give the answers to each question by explaining the practical application of each component and how that would shape their behavior. I.e. "As mentioned in the efforts heading, person #3 would theoritically be more inclined to complete the lego set faster as the process involves less creativity or inspiration and more effort." Explain, seperate, apply, question, answer. Obviously this is just one option of how you could do this. You may choose to not to and that's also fine. (I would recommend not linking the three components until H4 UNLESS you have enough words/not enough content in H3. Then you may want to make H4 a SH in H3. Again, this is totally up to you, it's just how i would focus on it.) H4 Creativity, inspiration, effort. Here is where I would tie in the three realms youve discussed and then loop it all back into 1. how people can generate more creativity, 2. how they can increase insipration, 3. how they can practice dicipline to increase effort application. Here I would really focus on the application of these constructs into practical senses. Maybe tie it into your first scenario. I.e. how the person applied motiviational psychology to increase their inclination to putting in effort which in turn improved their capacty to be creative as they did not believe themselves to be as limited in terms of skills as they has originally. You have 90% of your information already down pat, I just think it would benefit reorganisation to further clarify the flow of how your model works and tie it in a bit better to real world application. Good job and good luck [[User:Monuc9|Monuc9]] ([[User talk:Monuc9|discuss]] • [[Special:Contributions/Monuc9|contribs]]) 03:07, 9 October 2026 (UTC) ==General feedback== This looks like an interesting topic and I think the overall structure works well. one area that might be worth developing further is the effort -> inspiration side of the relationship. currently there is quite a lot explaining how inspiration can motivate effort, but less on how persistence and continued creative work could lead to new ideas. part of my research question is about goal persistence which is why this probably jumped out to me. A couple of sources that might be useful are: The dual pathway to creativity model: creative ideation as function of flexibility and persistence: https://doi.org/10.1080/10463281003765323 chapter five - the creation and curation of all things worthy: inspiration as vital force in persons and cultures: 10.1016/bs.adms.2020.01.002 - jack4234 <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an insufficient chapter # I don't think there was much inspiration or creative effort behind the drafting of this chapter; I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression; overuse of insufficiently acknowledged [[Motivation and emotion/Assessment/Using generative AI]]—express more in your own words #* integrate more strongly with other chapters by using embedded links to related topics <!-- Overall - Edit summaries --> #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is not appropriately acknowledged: #** Links to chat are not accessible #** No description of the changes made using genAI is provided in edit summaries <!-- Overall – Citations --> # Insufficient use of academic, peer-reviewed citations to support claims <!-- Overall – Word count --> # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FCreative_inspiration_and_effort&diff=2842768&oldid=2839542 these copyedits] |2= <!-- Overview – Comments... --> # Basic <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Reasonably engaging scenario or case study # The scenario reads like unacknowledged genAI content # Abbreviate the scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Use sentences and paragraphs # Description of problem is too long/overly complicated/generic genAI—explain the psychological problem or phenomenon in a simple way. Move detail into subsequent sections. # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are reasonably good # The focus questions could be improved by: ## there are probably too many overlapping questions # Are the focus questions based on genAI? |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Insufficient use of relevant psychological theory about this topic <!-- Theory – Builds on --> # Build more strongly on [[w:|Wikipedia]] articles and not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Insufficient use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # Provide more detail about the methodology and findings of key studies # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Basic integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Insufficient as a cohesive summary drawn from human reading of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # Address the focus questions <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Only partially counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ### This chapter "spins its wheels" by repeating the same ideas in different ways ## Bullet points are overused. Use full sentences and paragraphs. <!-- Written expression – Layout --> # Layout ## The chapter structure is underdeveloped; expand by using subheadings <!-- Written expression – Figures --> ## Figures ### Very well captioned <!-- Written expression – Figures - Citations --> ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – References --> ## The Reference is too brief ## References use very good APA style: ### Check and correct use of italicisation <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Add more embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Insufficient use of figure(s) # No use of table(s) # No use of feature boxes other than the opening scenario # Basic use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Basic use of the [[{{PAGENAME}}#See also|See also]] section ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use alphabetical order <!-- Learning features – External links --> # Basic use of the [[{{PAGENAME}}#External links|External links]] section ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order |9= <!-- Social contribution comments... --> # ~4 logged, useful, mostly moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:04, 27 October 2026 (UTC) 0mkujo5puaheles8v7kogli70y5spdq 1 332135 2844136 2843781 2026-11-03T08:35:28Z Jtneill 10242 2844136 wikitext text/x-wiki == Looking Good == Looking good, well done with all the theory used == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:UU3231448|Hi [[User:UU3231448|UU3231448]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:28, 18 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 1-level --> # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) <!-- Alignment with focus questions --> # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Reasonably good <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # Basic, relevant scenario; would benefit from further development to make it more compelling and illustrative # Add an image to the scenario to help attract reader interest <!-- Description --> # A promising description of the problem/topic is planned or presented # Focus more on the psychological theory (and evidence) that the author <!-- Focus questions --> <!-- Alignment and quality --> # Develop closer alignment between the sub-title, focus questions, and top-level headings # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |4= <!-- Key points--> <!-- Overall --> # Promising development # Insufficient use of citations # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- Other --> # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- GenAI ---> # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure -- # Relevant figure(s) presented <!-- Caption --> # Figure caption(s) should explain how the image connects to key points being made in the main text <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Insufficient # Only cite sources that you have consulted <!-- Systematic reviews --> # What are the top peer-reviewed theory and research journal articles about this topic? # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation |8= <!-- Resources --> <!-- See also --> # See also ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] <!-- External links --> # External links ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Basic <!-- Description about self --> # Very brief description about self – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:28, 18 September 2026 (UTC) == Explaining the systems == Hi, I found the job interview example helpful because it shows how different people can react differently to the same situation. I think the explanations of BAS, BIS and FFFS could be made even easier to understand by adding a small table comparing the three systems. For example, the table could show what each system responds to and how it might affect someone's behaviour. This would make it easier for readers to remember the differences between them. I also noticed a couple of small wording and punctuation errors in the introductory section that could be fixed when proofreading. I think the comparison table would be a useful addition for anyone learning about the theory for the first time. [[Special:Contributions/~2026-51041-43|~2026-51041-43]] ([[User talk:~2026-51041-43|talk]]) 12:24, 12 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an insufficient chapter mainly because it lacks a sufficient review and synthesis of relevant research # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic #* integrate more strongly with other chapters by using embedded links to related topics #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity <!-- Overall - Edit summaries --> #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FReinforcement_sensitivity_theory&diff=2843780&oldid=2841824 these copy edits] |2= <!-- Overview – Comments... --> # Good <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Description of problem is too long—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. <!-- Overview – Focus questions --> # The focus questions are good # The focus questions could be improved by: ## matching the top-level headings more closely ## splitting double-barrelled questions into separate questions ## focusing on how RST explains motivation and emotion (i.e., the topic) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Very good—key theories are well explained and applied; minor areas for improvements <!-- Theory – Builds on --> # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Reasonably good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # One figure and no tables or lists are used to help clearly convey key theoretical information <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Insufficient integration between the most relevant theory(ies) and the best research # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Basic summary and conclusion about RST theory # Insufficient as a cohesive summary of the best research about the topic # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # Address the focus questions <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is good <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. <!-- Style – Layout --> # Layout ## The chapter structure is problematic ## The headings could be more clearly aligned with the focus questions ## The chapter structure is underdeveloped ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## The sub-title was incorrect (fixed) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## The Overview and Conclusion should not have subheadings (fixed) ## Include an introductory paragraph before branching into sub-sections <!-- Style – APA style --> # APA style ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] <!-- Style – Figures --> ## Figures ### Limited use of figures ### Check and correct numbering ### Very well captioned ### Increase Figure size (fixed) <!-- Style – Figures - Citations --> ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Tables --> ## Table(s) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Style – Tables - Citations --> ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use reasonably good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) <!-- Style – References --> ## References use very good APA style: ### Check and correct use of italicisation |8= <!-- Learning features comments... --> # Basic use of learning features <!-- Learning features – Wikipedia embedded links --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Insufficient use of figure(s) # Basic use of table(s) # No use of feature boxes other than the opening scenario # Increase use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order <!-- Learning features – External links --> # Insufficient use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Add more links |9= <!-- Social contribution comments... --> # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:54, 1 November 2026 (UTC) osqk59tu4p94yyc11ijga8vmyys6n8l 2844140 2844136 2026-11-03T08:38:01Z Jtneill 10242 ## The number of references used is relatively low 2844140 wikitext text/x-wiki == Looking Good == Looking good, well done with all the theory used == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:UU3231448|Hi [[User:UU3231448|UU3231448]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:28, 18 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 1-level --> # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) <!-- Alignment with focus questions --> # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Reasonably good <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # Basic, relevant scenario; would benefit from further development to make it more compelling and illustrative # Add an image to the scenario to help attract reader interest <!-- Description --> # A promising description of the problem/topic is planned or presented # Focus more on the psychological theory (and evidence) that the author <!-- Focus questions --> <!-- Alignment and quality --> # Develop closer alignment between the sub-title, focus questions, and top-level headings # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |4= <!-- Key points--> <!-- Overall --> # Promising development # Insufficient use of citations # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- Other --> # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- GenAI ---> # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure -- # Relevant figure(s) presented <!-- Caption --> # Figure caption(s) should explain how the image connects to key points being made in the main text <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Insufficient # Only cite sources that you have consulted <!-- Systematic reviews --> # What are the top peer-reviewed theory and research journal articles about this topic? # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation |8= <!-- Resources --> <!-- See also --> # See also ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] <!-- External links --> # External links ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Basic <!-- Description about self --> # Very brief description about self – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:28, 18 September 2026 (UTC) == Explaining the systems == Hi, I found the job interview example helpful because it shows how different people can react differently to the same situation. I think the explanations of BAS, BIS and FFFS could be made even easier to understand by adding a small table comparing the three systems. For example, the table could show what each system responds to and how it might affect someone's behaviour. This would make it easier for readers to remember the differences between them. I also noticed a couple of small wording and punctuation errors in the introductory section that could be fixed when proofreading. I think the comparison table would be a useful addition for anyone learning about the theory for the first time. [[Special:Contributions/~2026-51041-43|~2026-51041-43]] ([[User talk:~2026-51041-43|talk]]) 12:24, 12 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an insufficient chapter mainly because it lacks a sufficient review and synthesis of relevant research # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic #* integrate more strongly with other chapters by using embedded links to related topics #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity <!-- Overall - Edit summaries --> #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FReinforcement_sensitivity_theory&diff=2843780&oldid=2841824 these copy edits] |2= <!-- Overview – Comments... --> # Good <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Description of problem is too long—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. <!-- Overview – Focus questions --> # The focus questions are good # The focus questions could be improved by: ## matching the top-level headings more closely ## splitting double-barrelled questions into separate questions ## focusing on how RST explains motivation and emotion (i.e., the topic) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Very good—key theories are well explained and applied; minor areas for improvements <!-- Theory – Builds on --> # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Reasonably good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # One figure and no tables or lists are used to help clearly convey key theoretical information <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Insufficient integration between the most relevant theory(ies) and the best research # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Basic summary and conclusion about RST theory # Insufficient as a cohesive summary of the best research about the topic # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # Address the focus questions <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is good <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. <!-- Style – Layout --> # Layout ## The chapter structure is problematic ## The headings could be more clearly aligned with the focus questions ## The chapter structure is underdeveloped ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## The sub-title was incorrect (fixed) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## The Overview and Conclusion should not have subheadings (fixed) ## Include an introductory paragraph before branching into sub-sections <!-- Style – APA style --> # APA style ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] <!-- Style – Figures --> ## Figures ### Limited use of figures ### Check and correct numbering ### Very well captioned ### Increase Figure size (fixed) <!-- Style – Figures - Citations --> ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Tables --> ## Table(s) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Style – Tables - Citations --> ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use reasonably good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) <!-- Style – References --> ## The number of references used is relatively low ## References use very good APA style: ### Check and correct use of italicisation |8= <!-- Learning features comments... --> # Basic use of learning features <!-- Learning features – Wikipedia embedded links --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Insufficient use of figure(s) # Basic use of table(s) # No use of feature boxes other than the opening scenario # Increase use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order <!-- Learning features – External links --> # Insufficient use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Add more links |9= <!-- Social contribution comments... --> # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:54, 1 November 2026 (UTC) o4yva5r5uk128u1cm7122zyo5l2qkxu 2844143 2844140 2026-11-03T08:58:51Z Jtneill 10242 Minor additions 2844143 wikitext text/x-wiki == Looking Good == Looking good, well done with all the theory used == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:UU3231448|Hi [[User:UU3231448|UU3231448]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:28, 18 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and/or subtitle not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 1-level --> # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) <!-- Alignment with focus questions --> # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Reasonably good <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # Basic, relevant scenario; would benefit from further development to make it more compelling and illustrative # Add an image to the scenario to help attract reader interest <!-- Description --> # A promising description of the problem/topic is planned or presented # Focus more on the psychological theory (and evidence) that the author <!-- Focus questions --> <!-- Alignment and quality --> # Develop closer alignment between the sub-title, focus questions, and top-level headings # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |4= <!-- Key points--> <!-- Overall --> # Promising development # Insufficient use of citations # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- Other --> # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- GenAI ---> # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure -- # Relevant figure(s) presented <!-- Caption --> # Figure caption(s) should explain how the image connects to key points being made in the main text <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using [[Motivation and emotion/Wikiversity/Tables|table(s)]] to summarise key information |7= <!-- References --> <!-- Overall --> # Insufficient # Only cite sources that you have consulted <!-- Systematic reviews --> # What are the top peer-reviewed theory and research journal articles about this topic? # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation |8= <!-- Resources --> <!-- See also --> # See also ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] <!-- External links --> # External links ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Basic <!-- Description about self --> # Very brief description about self – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:28, 18 September 2026 (UTC) == Explaining the systems == Hi, I found the job interview example helpful because it shows how different people can react differently to the same situation. I think the explanations of BAS, BIS and FFFS could be made even easier to understand by adding a small table comparing the three systems. For example, the table could show what each system responds to and how it might affect someone's behaviour. This would make it easier for readers to remember the differences between them. I also noticed a couple of small wording and punctuation errors in the introductory section that could be fixed when proofreading. I think the comparison table would be a useful addition for anyone learning about the theory for the first time. [[Special:Contributions/~2026-51041-43|~2026-51041-43]] ([[User talk:~2026-51041-43|talk]]) 12:24, 12 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an insufficient chapter mainly because it lacks a sufficient review and synthesis of relevant research # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic #* integrate more strongly with other chapters by using embedded links to related topics #* There is likely unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|use of genAI]]; if so, this violates principles of academic integrity <!-- Overall - Edit summaries --> #* Very good use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to communicate the changes made in most or all of the main edits <!-- Overall – Word count --> # ~1,000 words under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand <!-- Overall – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FReinforcement_sensitivity_theory&diff=2843780&oldid=2841824 these copy edits] |2= <!-- Overview – Comments... --> # Good <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Description of problem is too long—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. <!-- Overview – Focus questions --> # The focus questions are good # The focus questions could be improved by: ## matching the top-level headings more closely ## splitting double-barrelled questions into separate questions ## focusing on how RST explains motivation and emotion (i.e., the topic) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Very good—key theories are well explained and applied; minor areas for improvements <!-- Theory – Builds on --> # Doesn't build on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Reasonably good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # One figure and no tables or lists are used to help clearly convey key theoretical information <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Insufficient integration between the most relevant theory(ies) and the best research # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Basic summary and conclusion about RST theory # Insufficient as a cohesive summary of the best research about the topic # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # Address the focus questions <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is good <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. <!-- Style – Layout --> # Layout ## The chapter structure is problematic ## The headings could be more clearly aligned with the focus questions ## The chapter structure is underdeveloped ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## The sub-title was incorrect (fixed) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## The Overview and Conclusion should not have subheadings (fixed) ## Include an introductory paragraph before branching into sub-sections <!-- Style – APA style --> # APA style ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] <!-- Style – Figures --> ## Figures ### Limited use of figures ### Check and correct numbering ### Very well captioned ### Increase Figure size (fixed) <!-- Style – Figures - Citations --> ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Style – Tables --> ## Table(s) ### Add APA style caption(s) ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Style – Tables - Citations --> ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Style – Citations --> # Citations use reasonably good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) <!-- Style – References --> * More references are recommended: The number of references is relatively low. Incorporate additional relevant, high-quality academic sources to strengthen the evidence base and support key claims for a more comprehensive discussion of the topic. ## References use very good APA style: ### Check and correct use of italicisation |8= <!-- Learning features comments... --> # Basic use of learning features <!-- Learning features – Wikipedia embedded links --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Insufficient use of figure(s) # Basic use of table(s) # No use of feature boxes other than the opening scenario # Increase use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order <!-- Learning features – External links --> # Insufficient use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Add more links |9= <!-- Social contribution comments... --> # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:54, 1 November 2026 (UTC) o5lkiinbn2uum87eqspmjicleehf4mx 0 332412 2844051 2843806 2026-11-02T22:58:57Z Scogdill 1331941 2844051 wikitext text/x-wiki =Event= International championship boxing match between British Jem Smith and Australian Frank Slavin on 23 December 1889 in Bruges, Belgium. ==Overview== On 23 December 1889 a championship boxing match in Bruges ended with cheating on the British side: British Jim Smith was losing, so his supporters entered the ring, took advantage of Smith's holding Australian Frank Slavin and beat him up, using fists, sticks, "knuckle-dusters," "life preservers" and boots. The referee ruled the match a draw to prevent Slavin from being killed and then fled. The ''Cork Constitution'' called it "the Smith-Slavin piece of brutality."<ref>"Mr. Abington." ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&page=7&article=099&stringtohighlight=mandeville+viscount. Same title and p.</ref> A knuckle-duster was "A metal instrument made to cover the knuckles,"<ref>“Knuckle-duster, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  December 2024, https://doi.org/10.1093/OED/6130372797.</ref> like brass knuckles, and a life preserver was "A stick or truncheon weighted at one end" with lead.<ref>“Life-preserver, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  September 2026, https://doi.org/10.1093/OED/1055046410.</ref> One meeting of the boxing-oriented London Pelican Club was held almost immediately. Then a series of meetings following an investigation occurred in the first week of January (the [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club meetings]] referred to below). Slavin followed this fight with a challenge for American John L. Sullivan, saying he would fight in America or Australia, but not in a private venue in England. Many, many newspapers covered this fight and the reaction to it in the boxing community. Copy-and-paste journalism and the lack of copyright laws means that many of the stories are all or in part identical to others, so no attempt has been made to offer complete coverage. ==Logistics== * 23 December 1889, 8:00 a.m. * Stakes: £1,000 ==Related Events== * [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club Meetings about the Smith-Slavin match]] * the "[[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]]" ==Who Was Present== # [[Social Victorians/People/Mandeville|Kim Montagu, Viscount Mandeville]] # Jem Smith, British boxer # Frank Slavin, Australian boxer # Lord Rosslyn # Sir M. Sandys # Mr. G. Abington Baird, Smith's manager? and horseman (see description in Anthology) ==Entertainment== ==Anthology== The Sportsman The Tuesday, 24 December 1889 issue of ''The Sportsman'' devoted page 3 (7 columns) entirely to this story. This reporter's writing style is very ornate, and he also uses slang and local vocabulary: sabots, Brums, Newspaper Title: The Sportsman 24 December 1889, Tuesday: 3 [of 4], Col. 1a–7c [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891224&page=3&article=057&stringtohighlight=mandeville+viscount. Same print title and p. THE GREAT FIGHT. SMITH & SLAVIN FOR £1000. DISORDERLY PROCEEDINGS. BRUTAL ATTACK ON SLAVIN. SMITH KNOCKED DOWN REPEATEDLY. HE LEAVES THE RING. A DRAW DECLARED AFTER 14 ROUNDS. The long-talked-of fight between Jem Smith, Champion of England, and Frank P. Slavin, Champion of Australia, took place between eight and nine o'clock yesterday morning in the private grounds of a gentleman's residence near Bruges, Belgium. The contest, as everybody knows, was for 1000''l''. Thanks to the most cowardly and disgraceful behaviour of a gang of ruffians in Smith's corner, the referee was forced to declare the fight a draw at the close of the fourteenth round. Full details are given below. THE JOURNEY TO THE CONTINENT. (By our special reporter.) [?] travellers accustomed to make long and frequent journeys do not trouble themselves much about starting. The pangs of parting are easly effaced, and after a few years of "globe-trotting," as the Yankees are pleased to call it, beyond looking after personal comforts the travelling Briton has very little trouble. I have made many notable journeys upon all sorts of errands, but must confess that entering upon my last jaunt brought not a few  qualms — other than those likely to be experienced at sea. The preliminaries of the international fight, which was the reason of the trip from town, had not been carried through without many untoward circumstances arising to spoil the fun. On Saturday, however, there was every prospect of the business being brought to a successful termination, and when word was brought me that on the following morning our party were to leave London at an early hour there was no need for further hesitation, and preparations for the journey were rapidly made. What a day it was! Rain of a most annoying and penetrating kind fell until well into the night, making every one hurry to bring their day's business, or pleasure, to an end, that they might get indoors out of the dismal, depressing weather. Just as I had thought all safe for the [night came] A Rude Shock. [Arriving??] at the office of THE SPORTSMAN I found a well-known sporting journalist, whose expressive features at once conveyed to my practised eye that "something had happened." His right hand convulsively clutched a telegram, and it was quickly patent that this innocent-looking piece of paper was the cause of the trouble. "What do you think of this?" asked he, handing over the document. Only a few words were pencilled thereon, but these were enough to cause us alarm. "Stoppage, but think can get bail. — Slavin." Here was a pretty "go," the mystic words being translated meaning that the Australian had been "pinched" at Margate. Before we had thoroughly grasped the situation a second wire arrived containing the definite information that Slavin had been arrested at the Kentish watering-place and that he had a difficulty in finding bail among strangers. This dire news sent our spirits down to zero. Two acts of the little drama we were enacting had been very depressing, but the third brought sunshine again. Close upon the heels of the second telegram came a third, and it was with trembling fingers that at the request of my friend I opened the missive. A glance was enough — Slavin was free, and on his way to Dover. This sent us home satisfied that all would be well on the morrow now that both the principal actors in the contest were on the spot or nearby. Daylight brought with it A May Morning. Long before the earliest purveyor of the lacteal fluid arrived in the quiet suburban street to wake the sleepers with his hideous yelping, a cross between that of a beaten dog and a Comanche Indian, I had turned out, as there was no time to be lost. It was a delightful morning, the rain clouds having vanished, and being replaced by brilliant sunshine the weather being more akin to early summer than that usually associated with late December. The early promise did not last, as before eight o'clock the south-westerly breeze brought up rain clouds once more, and a fine free sample of "London peculiar" had taken possession of the streets as I wended my way towards Cannon-street Station. Ten o'clock was the hour named, but half an hour before that time several of the party had arrived at the trysting place. First to be noticed was Bob Habbijam, in company with two less-known sports, and this trio had not the dim terminus long to themselves. Jack Lewis, looking "all broke up," put in an appearance a few minutes later, while at the same time three scribes with pockets bulging from a surplusage of note-books dropped in. It was very soon evident that the company would be much larger than intended, as cabs rattled up continually, bringing a further complement. Tom Smith, Bob Kirby, Barney Sheppard, "Pop," and Frank Hinde were the next to join us, and in quick succession came Teddy Bayley, Will Riley, and Joe Vezey, who was to occupy the unthankful office of referee. According to the original conditions, a number of the intending spectators should have proceeded by way of Calais overnight, but by some means or other they managed to miss the train, and it was owing to this fact that so many were present on Sunday morning. There was no disguising the fact that such a crowd would draw attention, and militate against the secrecy necessary to insure us against intervention. There was no help for it, however, and before the Continental express came in from Charing-Cross "The Squire," Ike White, J. Kilby, C. Horton, C. Parsons, and J. Fleming had swelled the ranks of the ''voyageurs''. Down to Dover. It was only by judicious squeezing and packing that room could be found for all in the train, already well freighted with passengers, but eventually everyone managed to find a corner, luggage being conspicuous by its absence. I had a real sportsman for companion, Jack Kilby, of the Ferry Hotel, Cookham. Having missed the train overnight, not to be thwarted of his promised journey he arranged to proceed to town by road in the early morning. Instructing two lads to sit up through the night so as to be awakened in good time, Kilby turned in. At four o'clock he had the horse put in the shafts of a dog-cart, and drove slick away to Cannon-street. It was a particularly quiet trip, broken only by stoppages at Ashford and Folkestone, where some of the irrepressibles saw fit to leave the train and perambulate the platform. This, to say the least of it, was a foolish proceeding, as it fairly "gave the game away," and there was almost an air of expectancy about the officials when, after a run of nearly two hours, the South-Eastern express pulled up on Dover [Col. 1c–2a] Pier. Thoughts of coming trouble had ere this sobered down the playful ones, and it did not add to their joy when, looking seawards, they saw what is known as "dirty" weather in store. On the Briny. The transfer of passengers from the pier to the deck of the Princess Henriette was rapidly effected, and "once on board the lugger" my fellow-travellers disappeared as if by magic, but as rapidly came to light again, having, in view of developments likely to appear later, made arrangements below. There was a general changing of headgear, the bowler being discarded for the more useful tight-fitting cap — and what a variety were donned! As on the appearance of the clown in the pantomime, now the fun commenced. Before noon, when the hawsers were cast off, several shining lights in the boxing world prepared for the worst, the long rollers which broke in against the pier giving them the greatest alarm. Many had never previously ventured upon anything of a more stormy character than the Thames, and but for the fear of reproach would have turned back. As the "Princess" swung off from the pier the two "Bobs," Kirby and Habbijam, betook themselves to the "rail" in expectation of an attack of ''mal de mer''. What a pitiable sight they presented, but in a few minutes they found many brothers in misfortune, Tom Smith caving in. For a time it was a bit rough, and one of the good sailors commenced quoting Byron. "O'er the glad waters of the dark blue sea," sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season. Once away from the shore-swell it was smooth enough, and the sickly division came round wonderfully. Frank Hinde, who played his old part of low comedian, amused them, and at length by his wonderful flow of spirits — animal and otherwise — had the worst cases on their legs again. A few exceptions there were, half a dozen or so of the swells having "sought the seclusion which the cabin grants," and there they stayed. Our low-comedy merchant now found a a [sic] serious rival in "Pop," and between them they kept the coterie on the quarter-deck from being dull. Yarns were spun by this pair of drolls, and the merry jest went round until more substantial fare was in request at the table. Both the aforementioned did their share, the way Hinde put away the humble yet satisfying cold meat and pickles causing intense envy in the breasts of the less hardy mariners. A very quick passage was made across the streak, and at half-past three the coast appeared in sight. At the same time rain commenced to fall, and as we steamed past the lighthouse on Ostend breakwater it came pelting down sharply. At a quarter to three the good ship was secured alongside the wharf, and out came the lost sheep looking white and miserable, one passenger, although not of us, who had sat on deck more dead than alive the whole way over, fairly dancing with joy at his release from the purgatory of sea-sickness. Ostend to Bruges. While those unfortunate enough to be hampered with luggage stood by for the purpose of having it searched the rest of us secured seats in the train already drawn up alongside. It was then discovered that we did not start for half an hour, and the interval was turned to account by a  visit to the buffet. ''Cafe-au-lait'' was the general order, and mightily refreshing it proved to be. Everything comes to him who waits, but before the journey was continued sharp to the time mentioned, it was very patent that our descent upon Belgium in such numbers had attracted much attention. One dark-bearded "foreigner" who had watched us closely on the way across, actually grew so confident that he had exploited our movents that he grew bold beyond discretion. Coming forward with all the effrontery imaginable he rather startled us by asking, "Well, when is the 'mill' coming off?" In a twinkling one and all assumed a look of stony indifference, and on the question being repeated he was forcibly yet politely informed that we had nothing to do with any "mills." As we left the station our inquisitive friend was seen gesticulating to a belaced Ostender, evidently proud of his perspicuity. Thirty minutes' run through the flat and thoroughly soaked country brought the gable roofs of ancient Bruges in sight. Night had not yet darkened the streets, but there was mist enough to hide our movements to a great extent. It had been agreed that the brigade should split up as much as possible and spread themselves over the city. This was done in a great measure, Tom Dunning piloting my section across the  broad station-square to the Hotel de l'Univers. On the threshold I met Mick Taylor, who naturally looked somewhat aghast at the advent of so big a muster. It must have surprised him still more when others strolled in, and eventually the Britisher's party reached no less than fifteen. Fortunately our worthy host had plenty of available room, and after climbing up the winding stairs the whole troupe were comfortably housed. Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring that useful saponaceous compound with them. Entering the saloon after an outward and inward refresher, I ran across Josh Cosnett, and close behind the Brum stood, Frank Slavin, looming fine and large. The Australian Champion looked in grand condition, his face bronzed by the sea air and the toughening preparation he uses. Gently puffing a cloud from his Belgian weed, he greeted the arrivals cheerily, and was as skittish as a kitten, his hearty laugh resounding through the establishment. He had a very favourable passage across Channel the previous night, and amused himself by strolling around the city during the day. It was a piece of good fortune that not only could the ''garçon'' speak English fairly well, but at least two of my companions were conversant with the language of the country; negotiations anent dinner were therefore carried out with celerity. Pugilism at the Piano. While awaiting the refection Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys coon gave evidence that he was no mean performer on that instrument. Waltz, polka, and march all came as one to him. What a staggerer this would have been to the old timers, who never dared to allow a man in training such liberties. There was not the slightest sign of fear for the result of his coming encounter, and, in fact, Slavin appeared the least interested of the crowd. His piano selection whiled away the interval before dinner, at which the Australian presided. Taking his place at the head of the table Slavin entered into the fun, and no one would have conceived for a single moment that he had such a weighty matter on his mind as a fight. The repast was a lingering affair, and Slavin retired half way over the course, and betook himself once more to his favourite piano, which he wrestled with on and off during the remainder of the evening. While we were enjoying the good things under the roof of the Hotel Univers the rain had increased, and a perfect downpour succeeded. This had the effect of keeping the majority confined to the house, but several of the more venturesome spirits proceeded to explore the city, and they at any rate got a good drenching for their pains. The "pelting of the pitiless storm" continued the greater part of the night. Arrival of Jem Smith. The Champion of England, accompanied by Jack Baldock, Jack Harper, Jem Carney, and several other henchmen, came on from Lille at six o'clock, and arrived at Bruges about 8.30, taking up their quarters at the Hotel de Londres. Jem Smith, after leaving London on Friday, had travelled by easy stages to the Belgian frontier, so as not to miss a meal or proper night's rest and exercise between. He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training. Unlike his opponent, Smith kept very quiet during the evening, and retired early. A meeting of the representatives of the men took place at a late hour, and in company with the owner of the ground and others interested, the final arnuigements were made. I was present to assist at this necessary proceeding, and so perfectly had all the preliminaries been settled that there was very little left to be done[.] Owing to the presence of a mob of Birmingham men, who by some means or other had made their way to Bruges, there was fear of interference, but every precaution was taken to defeat anything disagreeable occurring. The behaviour of many in the streets during the evening was not calculated to assist those who desired to see a fair and square battle. Strict orders were given to the ''garçon'' to call us at 5.30 sharp, and to make assurance doubly sure many affixed their names on bedroom doors. Punctually to the moment came the rat-tat on the panel, and a hasty toilet was made by the light of the dip. A glance through the window reassured us with respect to the weather, the rain having ceased, but it was yet too dark to get any idea of the future. Even thus early the patter of sabots on the stones outside betokened the fact that Bruges, sleepy as it is said to be, was well awake. When the sounds from the famous belfry announced six o'clock [Col. 2c–3a] the bulk of the dwellers in the Hotel Univers had found their way to the ''salle-à-manger'', or "the manger" as "Pop" expressively named that apartment. Tea, coffee, fillets de bœuf, and chops were discussed, and the early birds had only been fairly started when with a cheery "Good morning, gentlemen," Slavin appeared. He lost no time, and was soon busy with knife and fork. Time flew by rapidly, and there were signs of an early daybreak. This hurried on the final preparations. Bills were paid, creature comforts taken on board, Slavin, with the greatest ''sangfroid'', seeing to his own business as unconcerned as possible. Our pilot put in an appearance shortly after the half-hour had chimed, and informed us that the other sections had already gone on. The March to the Battlefield. Breaking into groups of half a dozen the advance guard, led by an English resident in Bruges, pushed on, but before going two hundred yards there came a check. The railway had to be crossed, and just as the leading division reached the line they found the gates closed. Although opened in a couple of minutes it allowed the party to amalgamate, but they soon filed out agent. Many streets had to be traversed, and in the now increasing light we were scanned by groups of workmen and workwomen who were on the way to the many factories in the neighborhood. Talking except in whispers was tabooed, so that our nationality might not be discovered, and without further incident a walk of twenty minutes or so brought us to the chosen spot. Entering the private grounds by a wicket-gate, there was light enough to see that the others were present standing among the trees. Four moving objects suddenly appeared as we crossed the garden carrying some white posts, which quite gave Joe Vesey a fright. The principals were hurried off to comfortable quarters near by, Smith having the stable to make his toilet in, while Slavin was accommodated in a barn. Proceeding towards a big plantation I found several quietly at work Building the Ring. Between two belts of trees, formed chiefly of firs and beeches, was a clearing forty feet wide, and here the turf had been swept clear of the fallen leaves — in fact, a native, whose gigantic sabots caused much amusement, was still busy with rake and roller[.] Barney Sheppard superintended the pitching of the ring, an operation which took some time. Eight heavy stakes had been provided and plenty of rope, but the ground was very soft after so much rain, and wedges and guy-ropes had to be freely utilised. There being no tape, the distance was measured out by strides, and when eventually the job was finished it was plain that plenty of room had been allowed. The spectators amused themselves in various ways until, when the clock struck eight, all was in readiness. It was now broad daylight, and as several rustics on their way to employment caught sight of the, to them, peculiar proceeding, and stopped to more closely examine, the place did not promise to be free from interruption. All went well, however, and as the business did not concern them they passed on. There was some delay at this point, and it was close upon half-past eight when the men were reported ready for the fray. The following comprised the company round the ring: — Lord Mandeville, Lord Royston, Hon. M. Sandys, Mr Abington, Mr F. Kotchie, Mr J. Kilby, Mr C. Parsons, Mr Ike White, Mr W. Riley, Mr C. Morton, Mr Harris, Mr E. Bayley, Mr J. Lewis (Australia), Mr J. Vesey, Mr F. Hinde, Mr J. Fleming, Mr M. Taylor, Mr J. Calder, Mr G. T. Dunning, "Pop" Joy, R. Habbijam, G. ("Barney") Sheppard, Jem Carney, Tom Smith, Mangie [?] Hyams, Matthews, Summers, Boulton, and several others, including two or three reporters. Smith, accompanied by his seconds, J. Harper and J. Baldock, entered the ring, quietly going over to the south-west corner with hardly any notice. Baldock at once wanted to know why Slavin had not turned up, and was inclined to be nasty, when the Australian walked briskly to the opposite corner, having, without the customary teasing for choice of positions, allowed Smith to take the higher ground. Josh Cosnett and Bob Kirby accompanied the Australian Champion as seconds, while Mick Taylor superintended the arrangements in his corner. A loud cheer greeted their advent. There was a few minutes' delay after the men had taken up positions, owing to Slavin not being provided with a seat, and Baldock again wished to hurry him up, but matters proceeded peacefully. Mr Joe Vesey, who had been appointed to the unthankful post of umpire, took up a position on the lower side midway between the corners, and Mr J. Calder was selected as timekeeper. On either side of the referee Mr J. Fleming, umpired for Smith, and Mr W. Riley for Slavin, these preliminaries being quickly settled. The disposition of the officials made Slavin's corner appear the stronger, and to still further add to this, a large percentage of the spectators selected that side, so as to get their backs to the wind. Another, and, perhaps, a more potent reason was the fact that The Dreaded Birmingham Division had massed themselves behind Smith and along the higher aide. They had long caused considerable fear among the spectators, but were not so noticeable until they stationed themselves in a phalanx near the Englishman. While the corners were being arranged several of the "boys" worked their way round to Slavin's quarters and stood menacingly by. Mick Taylor was too old a bird to be caught, and with the assistance of Prank Hinde, Lord Mandeville, and one or two trusty friends, guard was kept over the water and other articles. Every necessary for the two athletes had been provided by the proprietor of the grounds, so that there was no trouble on this score. Smith was wrapped in a heavy frieze overcoat, while Slavin, who, during his sojourn in the barn had been divested of his heavy moustache — a loss he deeply regretted — was also covered by a big ulster. It was now half an hour beyond the time fixed for the commencement of hostilities and broad daylight, even the belfry of Bruges being plainly visible in the distance, but curiously enough not a soul beyond the pugilistic party was anywhere near the fighting-ground. At last everything was ''en règle'', and in perfect silence Mr Joe Vesey came forward and said, "I hope all those present will keep quiet and help me all you can," at which brief speech there was a chorus of "hear, hear," especially from Slavin's corner. At 8.35 Belgian time, the two antagonists met in the centre of the arena and shook hands, the seconds, as in time-honoured fashion, doing likewise, forming what is known as the Maltese Cross. Smith, who partook of a draught before commencing, was the first to peel, and although he looked fine enough in the face, when his covering had been removed he appeared fat, although nothing like so obese as when he met Peter Jackson. The Englishman wore short white drawers and green socks, while a strengthening plaister adorned his waist. Slavin looked the picture of health and strength, having trained down very hard, his muscles standing out most prominently. He was clothed in long black drawers and blue stockings, and, like his opponent, wore a deep plaister. In addition Slavin used a suspensory belt loaned to him by Charlie Mitchell, who had found it of great service in his fight with John L. Sullivan. Under weather, which for the business in hand was simply perfect, and particularly warm for the time of year, there was, as far as could be seen by the uninitiated, every prospect of a good [Col. 3c–4a] fight. This was doubted by the majority, the presence of the crowd of roughs betokening trouble when once the battle should commence. The Call of "Time" At twenty-five minutes to nine brought everyone to the ropes, and amid perfect silence the pair of pugilists shaped for the encounter. Slavin from the first smiled confidently, while Smith looked serious. Neither appeared in any hurry to begin, and sparring for an opening, Slavin gradually worked Smith back into his corner. Fifty seconds elapsed before a blow was struck, but this had not passed without our opinion being confirmed, that we might expect trouble from the Brums. Smith's seconds set the ball rolling by hurling the vilest of epithets at the Australian, and their example was followed by their supporters. By some oversight the seconds had not been ordered out of the ring, but attention was at once called to the fact, but, constituting himself referee, Baldock refused to obey the order to retire when given by the appointed officials. This boded ill, and before a round had been fought it was pretty plain to all that Slavin would have little chance of victory. Had the objectionable party confined themselves to vituperation this might have been borne, but going from bad to worse we soon witnessed a scene that was not only a blot upon the good name of Englishmen as lovers of fair play but one never equalled even in the worst days of the ring. Slavin was the first to lead, but only tapped his opponent lightly on the face, and he had the best of several exchanges, but did not put forth all his power into his left-handers. Smith's blows were also comparatively harmless, until the last half minute, when they got to close quarters and hammered away sharply amidst a chorus of yells from the Smithites, until they fell together. No sooner were the two men down than Baldock rushed up, and in picking his man up knelt on Slavin. A foul was claimed, but no notice was taken of it, although it was very palpable. Warnings were given to the Australian not to go into Smith's corner, but without avail, and he entered upon the second round full of pluck and confidence. Language — if such filthy expressions as were levelled at Slavin could be so designated — of the violent kind greeted him on facing Smith, and this was not confined to the hired mob. Those from whom at least fair play might reasonably have been expected indulged in blasphemy and rowdyism even more pronounced than that of the others. To still further disconcert Slavin a claim of first blood was made. The moment he walked across the ring Smith worked up close at once, and on rushing in was met with the left on the mouth. A brief struggle ensued before they fell together, Smith while on the ground spiking Slavin in the thigh. A foul was again claimed but the referee was so badgered that he could hardly have known where he was, and no notice was taken of the occurrence. In the third round, again a short one, Smith was Knocked Down By a cross on the jaw, and he came up looking the worse for wear next time, while the blackguards in his corner were more threatening than before. Smith again waited in his corner, and, nothing daunted by their menacing attitude, Slavin boldly went after him. As he advanced he was assailed by the miscreants, who now saw that Slavin was getting the best of it, and when after another round had been fought, in which Smith began to blow freely, the Australian was near the ropes he received a blow from one of their number. Every time Slavin was forced to fetch Smith out of his corner, where at this early stage the ropes had been loosened. In the sixth round Slavin now warmed up, sent home a hot left-hander on the right eye, from which the ruby fluid ran freely. This he followed up by some sharp work, eventually fetching Smith down with a vicious right-hander. This treatment of their "Champion" did not suit the villanious [sic] coterie, and Slavin went across the ring next time at great risk. Smith worked Slavin round to the ropes, and while there the dastards held him for the Englishman — save the mark! — to punish, and there was an indescribable scene of confusion. Still, Slavin would not be denied, and in the eighth round sent the right heavily on the jaw, and fairly knocked Smith off his legs. In the following round Smith received a smash on the mouth, but Slavin was terribly handled by the outsiders, who punched him freely while in holds. Smith up to this had hardly made a mark on his antagonist, and beyond a cut on the lip he showed no signs of punishment, but the cowardly ruffians who assisted the "Champion" wounded him severely. So vile was the conduct that the referee was, like every one else had long been, thoroughly disgusted, and he tried without avail to put a stop to it, calling upon Harper to "be a man." Smith was generally short in his deliveries, and Slavin, paying more attention to the body had doled out some telling work. In the twelfth round Smith was felled in his corner, and this brought a crisis. The latter broke away from Slavin's attempts to get on with the left until he was almost backed up against the corner stake, when the Australian dashed the left on the face. No sooner was he fibbing away than the ruffians crowded up and showered blows on his head. In the midst of the mélée a cry of "Here they are" was raised, and A Stampede Ensued from Smith's side. It was evidently a concerted plan, as when the leaders found that no one stirred from Slavin's corner they actually came back, and pointing to the roadway, yelled, "There they are, there they are." By this time Smith had run a hundred yards away towards the house, and Slavin's party, under Rule 26 [?] claimed the fight, but to no purpose. The confusion ensuing upon this manoeuvre looked like putting an end to further fighting. Slavin, who had shown not only admirable pluck and judgment, but also kept his temper, was quieted down when he thought there was yet a chance of winning. Smith was fetched back again, and time was called. Slavin, before putting his hands up, addressing his remarks to the crowd at the other side of the ring, said, "Are you Englishmen? I am an Englishman, and only want fair play." The fight then proceeded. Seeking to settle matters now that he had Smith weakened, Slavin went right over to Smith, and planted the left sharply on the body twice, and then dashed in with both bands, landing on the jaw. As they closed someone of the brutal mob struck Slavin on the right ear again with a "duster," the referee saying, "I saw that." As Slavin threw Smith and the mob again became aggressive, Vesey, who could stand it no longer, said, "I refuse to act any longer. It is impossible for this man — pointing to Slavin — to get fair play." The referee concluded by giving the decision, "It's a draw." The mob heard this with fiendish delight, and at once cleared off from the ring-side. Slavin could hardly contain himself, shouting "There is not a man round the ring can kill me. l came 16,000 miles for a fair fight, and why don't you give a man a chance." There was no help for it, and the latest fight for the Championship ended under circumstances which brought the greatest discredit upon those who were responsible for [Col. 4c–5a] the presence of the wretches, who were prepared for any villany [sic] to prevent Smith being defeated. Not satisfied with their performance at the ring-side, they followed him to his temporary dressing-room, but were soon persuaded to leave him alone, which was perhaps fortunate for at least some of the party. Jem Smith had a carriage in waiting for him, and was rapidly carried off to the town, where it was intended to stay for some hours. There was a rush to the telegraph-office to send messages to interested friends, but only a few minutes were available, as at least a third of  the party intended Making All Speed Homewards. The Ostend train was waiting in the handsome station, and there was no time to be lost[.] Slavin, who walked the whole of the way back into town, was hurried into a first-class compartment, where he made himself comfortable in a corner with a cigar. There was a crowd of hangers-on in and around the station, who evidently were by this time aware of the nature of the morning's work, but without further incident the train steamed out of Bruges. Arriving at Ostend at 10.15 we found the Comte de Flandres ready to receive passengers at the quay-side. This boat is not to be compared with the Princess Henriette, being much smaller and less seaworthy. Slavin had a deck cabin prepared, and therein held a levée, champagne and other sparking wines being ordered up. The Australian, under the careful hands of Mick Taylor, had the cruel wounds inflicted by the ruffians at the ring-aide attended to, and by the time the boat was clear of the offing Slavin was fixed up in shape and paraded the deck. Everyone talked of the fight and its terrible surroundings, and nothing but the severest condemnation was accorded the Englishman's party. One gentleman, who was wearing Smith's colours, was so disgusted that he threw them overboard, and replaced the kerchief with one bearing Slavin's monogram. Dover was reached in due course, a big crowd awaiting Slavin's arrival on the pier, but, shorn of his moustache, he was hardly recognised. He proceeded thence to Margate with M. Taylor and J. Cosnett, where another ovation awaited him. The other portion of our company journeyed direct to London, and on detraining at Cannon-street there was a crowd collected to gather news from the seat of war. Smith was to leave Bruges during the evening, and all being well will arrive in London early this (Tuesday) morning. THE FIGHT. First Round. Slavin left his corner quickly, with a firm, determined step. Smith was much slower in his movements, and the Australian had almost reached his opponent's corner when the men were face to face. The disparity in the height of the men was very marked, Slavin standing well over the Englishman. Each wore a deep strengthening plaister, whilst Slavin, it may also be added, made use of the suspender worn by Charley Mitchell in his memorable fight with John L. Sullivan. Full of confidence, and evidently in splendid condition, the colonial looked every inch a fighter, his length of reach, and lithe, agile movements contrasting strongly with the lumbering position struck by Smith. Though fairly clean in the face Smith looked very heavy below, and undoubtedly carried too much flesh for the business he had taken in hand. Great caution was exercised on both sides, and it was seen that neither man intended to throw the slightest chance away. Weaving in Slavin feinted with his left, but Smith was quickly out of danger, and in turn Slavin dodged an intended blow from the Englishman's left. In this manner fifty seconds sped on, after which Smith got in lightly on the month. Exchanges of a light character followed, Slavin reaching the body and Smith the face. Subsequently the latter, who seemed considerably puzzled by the Australian's style, several times led out of distance at the head, Slavin on each occasion breaking ground very prettily. At last the men got to close quarters, where some sharp infighting ensued. Slavin made the body his mark, where he put in heavy work with the right, Smith in the meantime visiting his opponent's head. The Australian's blows seemed to take the most effect, and the round ended by Slavin having the worst of a fall. Time, 2min 30sec. Second Round. No sooner had the men been removed to their corners than Jack Harper called out in a very loud voice, "First blood," a claim which was of course allowed, the Colonial showing a slight trickle from the lips. On "Time" being called the pair were quick to scratch, that is to say, Slavic walked right across to Smith's corner, the Englishman refusing to come out. A long opening spar took place, but at length blows were exchanged, nothing very serious being done, and closing, the pair had a slight struggle, both falling together. Whilst on the ground a good deal of kicking and struggimg was indulged in, and whether intentionally or no Smith spiked his opponent on the inside of the left thigh, tearing his drawers, and causing the blood to flow considerably. Uproar followed, the round lasting 20sec. Third Round. Though up to this time neither man had sustained much punishment, the exchanges not having been very severe, Smith was always carried to his corner, Slavin, on the other hand, walking beck to his. At the beginning of the third round the referee ordered the seconds out of the ring, but they declined to go, saying that official had no power to order them to do so. On starting Smith forced the fighting, and leading with the left, hit Slavin upon the body. This he followed up rapidly with two more in the same place, all light hits, and some rapid exchanges followed, the Colonial using the right with effect upon the side of the head. Smith paid principal attention to the body, but at length Slavin, drawing back the right, shot it out straight from the shoulder, catching the Englishman fair on the point, and sending him down all of a heap. This magnificent delivery put the Slavinites in excellent spirits, the Colonial smiling all over his face, while the Smithites looked serious. First knock down blow was claimed, and, of course, allowed for Slavin. Time of round, 24sec. Fourth Round. Smith rose from his seat looking very serious indeed, Slavin, on the other hand, being in much better spirits. At the same time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the Colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. The remainder of that party stayed in Smith's [Col. 5c–6a] corner. As for the seconds, all remained in the ring. The opponents sparred for the first twenty-five seconds, and at length Slavin, getting an opening, led out of distance at the body. The Englishman seemed glad to have a blow, but Slavin weaving in delivered lightly with the left upon the cheek. Again the pair faced each other in the centre of the ring, and Slavin went for his man, and feinting with the left, just failed to bring off the right, Smith skipping out of the way. Again the Antipodean feinted with the left, and following up the charge, this time successfully caught Smith flush on the mouth. Then quick as lightning getting to close quarters, Slavin delivered heavily right and left on the body, Smith being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell, the seconds, rushing to the centre of the ring, falling over them. Time, lmin 15sec. Fifth Round. "Don't go into his corner," urged Taylor and others of Slavin's party to their man, but nothing could restrain him when time was called, and, springing up with a bound, he advanced towards the opposite comer, where Smith was standing up. Once or twice the Englishman seemed inclined to go into the middle of the ring, but was always prevented by his yelling seconds, who stood on either side of him. On toeing the mark Slavin appeared with a suspicious-looking redness of the left eye, caused by the miscreant above referred to. Though appealed to the referee declined to give any decision on the subject, and the fight proceeded amidst some of the most horrible language it has ever been our misfortune to listen to. This proceeded entirely from Smith's corner, who, seeing that their man was already beginning to blow and look distressed, prepared to stop at nothing. Slavin, after wasting a second or two, followed Smith into his corner. One or two exchanges, quickly delivered, of a light nature took place, and the Colonial got in on the body with the right, following up with the same on the head. This Smith considered ample justification for sinking gracefully upon the ground, the round having lasted only sixteen seconds. Sixth Round. Smith came up blowing Iike a grampus, whilst on the other hand, Slavin, seeing his advantage, was eager to begin. Of course, it was necessary to go right over into Smith's corner, that worthy not moving two feet from his chair, whilst the shouts, jeers, and din which greeted the Australian were of an inexpressibly disgusting nature. Still, nothing daunted, the Antipodean followed his man up, Smith leading at the head. This Slavin easily turned aside by a lithe movement, and in turn was stopped by Smith. Resuming, Slavin banged in the left on the mouth, Smith countering on the body. A smart counter ensued, and at this juncture the referee entered the ring, making a fruitless appeal to Smith's corner to behave themselves. Nothing, however, could silence the foul language which proceeded from that quarter. Once more facing each other, the Australian shot in the left fist heavily on Smith's left eye, opening a severe cut, which bled freely. Some vicious infighting followed, and Smith twice got the right in upon the head. Slavin countered with a terrific right-handed blow on the eye, knocking Smith down. Time, 1min 20sec. Seventh Round. In vain did Slavin's friends implore him not to go into Smith's very "warm corner." It was plain that if he did not do so there would be no fight, Smith never coming many feet away from the rascals who had been hired to protect him. Getting to close quarters Smith delivered the left lightly upon the face and banged in directly afterwards the right on the body. Slavin did not see the Englishman having matters all his own way, and a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, whilst, naturally, Harper and Baldock were yelling at the top of their voices. The confusion wars something terrible, and the ring looked very much like being invaded. The ropes in Smith's comer were pulled right down, so that an entrance might be effected at any moment, and it was a relief when the pair wrestled and came down in a dog-fall together, the round having lasted one minute. Eighth Round. Crossing over at once to the only place where Smith was to be found (''i.e.'', in his own corner), Slavin worked his opponent round to the side of the ring, and there let him have the left lightly upon the face. Standing at this moment close up to the ring-side, the Colonial received a couple of smart blows from a stick which a ruffian coward carried.  An immediate cry went up, and the individual finding himself under very close observation immediately slunk behind. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's following him about and jeering in his face. Again the referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith, who was receiver-general, judged it advisable to seek mother earth.  Time of round, 1min 30sec. Ninth Round. This was altogether a very curious round. When time was called Slavin got up to find his opponent, who barely moved a foot from his chair, and this fact  should have been sufficient warning that something awkward was intended. In the corner the two sparred for about fifteen seconds amidst a horrible din. Finding that the Englishman was in no hurry to begin, Slavin made the first movement, and landed with the left lightly on the head. Following up this, he brought the right round heavily on the dial, making Smith wince, but a second blow in the same quarter was avoided. Still in the corner, the pair got to half-arm fighting, once more bringing them close to the ropes. The Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. The referee, though appealed to again and again, took no notice, though admitting that he saw what was going on. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely. This last blow sent Smith to grass, the round occupying 50sec. [Col. 6c–7a] Tenth Round. The Champion of England by no mean relished the treatment received in the last round, and at the expiration of thirty seconds was puffing and blowing freely. He attempted to spar for wind, but the Australian seeing his advantage bored in when he could find an opening. Delivering right and left, and receiving only slight returns from Smith, the Australian pulled his man over, and the round terminated in 40sec. Eleventh Round. Neither was quite so quick to time, and Harper, noticing that, said, "This is not a two-round job, Patsey." Something like fifteen seconds was spent in sparring. Following Smith up the Australian failed to get an opportunity for some time. At length he put his hands down and invited Smith to "Come on and have a round." This was evidently not a particularly pleasing invitation to the Champion, as he remained in his own corner. Not to be denied, Slavin went after his man, but a left-hander meant for the head was put aside by Smith. At the end of a minute Slavin landed heavily on the Englishman's left cheek, a great red blotch immediately appearing. A rapid exchange followed, and Slavin was again home on the body. A heavy blow on the mouth from the Antipodean sent his opponent to the turf, the round having lasted 1min 35sec. Twelfth Round. A great change was noticeable in the appearance of the rivals on meeting each other for this round. Smith was showing signs of punishment on the face to a very large extent, his left eye being cut, his lip split, and there being a number of marks of Slavin's handiwork on his face. He was very distressed, and blowing heavily. On the other hand, Slavin, beyond the bad left eye (not done by Smith, but by one of the gang of hirelings), had apparently nothing the matter with him. Springing up brightly to his work, the Australian roamed about the ring until he found Jem Smith secreted in his own corner, only just away from his chair. The Australian reached the mouth lightly, and endeavoured by this means to draw his opponent out, but failed to achieve his object, Smith being strenuously warned not to move away from his seconds. The corner was encroached upon by Smith's crew, Harper and Baldock this time being almost between the men. The tumult was very great, but it seemed impossible to daunt the Sydney man, who followed Smith up, though little harm was done. Sparring was the order of the day until Slavin worked in with the left, delivering heavily. With great rapidity he drove the right with terrific force right on to Jem's chin, knocking him clean off his pins, and ending the round. Time, 1min 30sec. Thirteenth Round. Smith came up thoroughly dazed, the terrible blow in the last round having taken a lot of the steam out of him. The Birmingham gang, seeing that this was the case, got all of a heap in Smith's corner, and though Slavin was most earnestly advised to keep away, he would not do so. Smith stood just by ''his'' chair, and going right up to him, Slavin let out with the left, and bringing round the right. Both of these visitations Smith missed, but not so in the case  of a second smack from the right, which made Jem wink. However, turning round, the Englishman forced his opponent up against the ropes, and here occurred one of the most disgraceful scenes ever witnessed at a pugilistic encounter. Some villain drove his knuckle-dustered fist right on to Slavin's ear, cutting it almost in two and causing the Australian to exclaim, "Oh!" Other of these demons kicked him, a scene of indescribable confusion ensuing, in the midst of which the men continued fighting until the pair came down together. Time. 1min 28sec. No sooner was the round finished than cries were raised of "Here they come: the coppers, the coppers; and all Smith's gang, including his seconds, rushed away from the ring, at the same time calling upon Slavin to do so likewise. As for the Champion of England, he headed the runaway party, though nobody could detect the shadow of a "blue." The Australian never left the ring, much to the chagrin of the Smith party, who, seeing that the referee was being called upon for a decision, returned rather sheepishly. Baldock was in a dreadful fright, thinking the battle has been lost, but when the words "fight on" came he and his friends were very delighted. Fourteenth and Last Round. When "Time" was called Slavin folded his arms and waited for Smith to "come on," but Mr Smith didn't want to come on, and remained in his corner. Seeing it was impossible to make him stir Slavin once more crossed over, and a spar of nearly a minute took place. Then Slavin drove home the right on the body heavily, and the pair closed in Smith's corner. This gave another opportunity to a miscreant to strike Slavin on the head. Others in the gang were not slow to follow, and again the round ended in horrible confusion. Time. 2min 15sec. No sooner was this round over than the referee called out. "I refuse to act any longer. It is impossible for Slavin to get fair play. It's a draw." Then, going up to Slavin he said, "I will not stay to see you killed." The Antipodean stood up and exclaimed, "If you are Englishmen why don't you give me fair play? I have not had fair play." After this came a stampede. Remarks. Judging by the disgraceful proceedings recorded above, we are surely near the end of what are known as "fights in the old  style." That such a scene as that witnessed yesterday could possibly be enacted by any man with a drop of Christian blood in his veins seems almost incredible. Take the picture. An athlete, one of our stock, journeys 16,000 miles to measure blades with the best we can produce. Confident he will receive fair play in a land which boasts of nothing foul, he, after trying in vain to ratify a match to meet England's Champion, to be decided before an impartial and unbiassed crowd, puts down his hard-earned hundreds and meets our man in the style the latter insists upon. And with what result? Battling under unheard-of difficulties, not fighting one man but nearly a dozen, he narrowly escapes being murdered, and in the end is robbed of a victory he had won not once but in almost every round contested. If the nail driven hard into the coffin of pugilism yesterday is not the last, it is indeed ten thousand pities. Nothing can flourish, however, without support, and, providing the remarks passed by the majority of the spectators at the ring-side at San Michael are of any account, it will be many a long day before a knuckle-fight receives the patronage of the "upper crust." Slavin, we say, narrowly escaped murder, a fact proved by the words the referee, who, as well as others present, stood in fear of his Iife. One gentleman, an old ring-goer, affirmed he had never witnessed such disgraceful proceedings, and when the ring was finally cleared a feeling of relief was manifest on all sides. Unfortunately we are called upon to say that the riot and murderous intent were all in Smith's corner. On the other side not a man was present against whom a breath of suspicion could be raised. Opposed to Slavin was a hired gang of desperadoes ready to commit any act necessary to serve the ends of those by whom they were engaged. Terms of imprisonment had been worked out by each, and, prepared with knuckle-dusters, revolvers, and knives, they appeared ready to butcher the company at the raising of a finger. In face of all this the Australian fought his way to a moral victory. Almost from the outset Smith was outmatched, and beaten at the game considered his own. Sticking to his corner, of course by order, he showed none of his boasted bull-dog pluck and courage. His dreadful body-blow was easily put aside by Slavin, whilst instead of being "bashed" on the ground and battered out of shape, the Australian fairly held his own in wrestling. We are not blaming Smith for the doings of the miscreants present, and we also know that he was acting under orders by staying in his corner and affording them every opportunity to kick, strike, and prod the Australian, but it was altogether un-English to hold Slavin on the ropes whilst the mob belaboured him with "dusters" and sticks. Smith was, besides being in poor condition, slow and ponderous. After the third round his chance was hopeless, and his supporters were surprised at the form shown by Slavin. Contrary to general expectation, the Cornstalk proved himself far and away a better knuckle-fighter than a "glover." With his long reach he did great execution by left-handed deliveries, and the way he toppled Smith's fourteen stone over with the terrible right was a sight [p. 3, Col. 7c – p. 4, Col. 1a] Page Number: 4 that would have gladdened the heart of old Bill Richardson. In a word Slavin was entirely a different man from the Slavin that fought Bill Goode. Brimful of confidence he faced dangers that few would have encountered, and had the battle been conducted on fair lines he would have been Champion of England and Australia in less than half an hour. In all his battles he has never known defeat nor met with a knock-down blow. Yesterday it was not once found necessary to carry him to his corner, and as all the fighting was done in Smith's corner he must have walked a matter of eighteen yards each round. Slavin has a method of his own in pugilism. Possessed of staying powers, or "bottom" of a most remarkable character, he tires his man and then fights him. Yesterday, providing he had met with no punishment from outsiders, he would have left the ring almost without a scratch, and, as he himself said at the finish, "he was just getting warm." Remarkably clever on his feet, Slavin was in and out with the activity of a feather-weight, and feinting with the left, which he brought forward, he sent his right under the ribs with crushing effect. For some time past it has been an open secret that if Smith did not win he would surely not lose, and it is a blot on the proverbial love of English fair play to think that a mob should have been organised, and such a cowardly attempt made to rob a stranger amongst us of a victory so hardly earned and so clearly deserved. From the first the rules were broken by allowing the seconds to be in the ring whilst the fight was in progress, and although Smith's men threw everything in the way of the Australian, Cosnett and Kirby never transgressed the rules of fair play. Why the referee did not award the fight to Slavin when Smith left the ring in the thirteenth round can only be explained by his fear of the consequences that might follow.  Bearing on this subject, Rule 26 says: — "That if a man leaves the ring, either to escape punishment or for any other purpose, without the permission of the referee, unless he is involuntarily forced out, he shall forfeit the battle." Rule 24 is also plain enough in all conscience, "That any man voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight." Undoubtedly Mr Vesey lacked firmness, and perceiving this at the start, the "boys" and Smith's seconds, not to mention another couple who used their hands and feet at every opportunity, saw the game was in their own hands, and simply did as they liked. From first to last the proceedings were of a most disgraceful character, and the result showed plainly the reason why Smith's side objected to all the referees nominated by Slavin. Several gentlemen present, in fact all disinterested spectators, expressed their disgust at the treatment meted out to Slavin, and it was suggested on the return journey that a subscription be opened to provide funds for a testimonial to the plucky but unfortunate Colonial, particulars of which appear elsewhere. Even after the fight Slavin's assailants could not let him rest. Not content with robbing him of victory, they followed him to his place of dressing, but a determined front shown by Mr Taylor, Cosnett, and Kirby kept the blackguards back. These are well-known, and will, we believe, shortly meet with the punishment they so richly deserve. One rough made a rush for Lord Mandeville just before the close, but "Kim" soon showed he could hold his own with the "boy," who beat a hasty retreat at the "sight" of six-inch "Jack." We now take our leave of this unsavoury affair, which will probably be the last of its kind — "a Championship knuckle fight." The Englishman's boasted love of fair dealing appears lost to the professional fighter, and the sooner "merry mills in the old style" die out, and the hanger-on has to earn his living in some other way, the better. Slavin will visit London today. He leaves Margate by the 12.15 train, and will arrive at T<small>HE</small> S<small>PORTSMAN</small> office about three o'clock. Smith had a nasty lump on the jaw and left cheek, whilst his right eye was cut open deeply. In addition to this the punishment he sustained on the body was of a severe character. Slavin, on the other hand, only received a cut lip until after the fourth round, when the rowdies joined forces with Smith and smashed his eye during a mélée. After all their kicking and the free use of "dusters" the Australian only suffered from a badly-cut ear and the before-mentioned damaged eye. It may be stated that not a single bet was made at the ring side. SLAVIN'S OPINION OF HIS TREATENT. His Future Plans. On the return journey to England yesterday Slavin was the centre of attraction. At Ostend, where the tale of the fight had somehow or another been told, he was met by an eager crowd, and on the boat he held quite a ''levée''. Though at first somewhat disappointed at being robbed of the title of Champion and 500''l''. so well within his grasp, the Australian, surrounded by old friends, and new ones gained by reason of his straightforward conduct, rapidly regained his spirits. With the inner man refreshed by a steak, washed down with some of Heidsiek's [?] best, Slavin, a Havana between his lips, quickly got on good terms with himself and the rest of the  party. Much sympathy was expressed on all sides, and in the course of conversation with a S<small>PORTSMAN</small> representative Slavin said:— Of course, I feel a bit cut up, because I had my man beaten[.] Did you expect any trouble at the ring side? Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. Do you feel any effects of the fighting? Slavin: The fighting? I feel nothing of what I got from anyone in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. Why did you persist in going to his corner? Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. Of course you heard a mob was got together for Smith? Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. What do you think of Smith? — Well, I think so much of him that I'll bet him 200''l.'' I stop him in six rounds, but we must have no more private shows. I've had enough of that business. I suppose you have a poor opinion of English fair play? Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. With the gloves? Slavin: No, I went to fight him with the knuckles. I think I should get fair play out there. You prefer the old style to the new? Slavin: I am fond of boxing, but the other is my game, and though not a boaster I fear no one — that is if I can have fair play. Shall you give any exhibitions? Slavin: I may perhaps make a tour round the country, and very likely visit Ireland. You put on weight the last week? Slavin: Yes, I gained eight pounds and felt grand; in fact, just up to dick when Smith bolted. What do you think of our seconds? Slavin: Well, I'd like them to have a job in Australia. They'd never come back again. When do you propose visiting London? Slavin: I'll come up to-morrow and call at T<small>HE</small> S<small>PORTSMAN</small> Office about three. A stroll was now taken on deck. [sic] and reaching Dover Slavin was met by some friends from Margate, who carried him off to the train in triumph. A Testimonial for Slavin. So impressed were some noblemen and gentlemen with the pluck and ability shown yesterday by Slavin, and at the same time thoroughly disgusted at the treatment he had received, that it was decided whilst crossing the Channel on the way home to open a subscription on his behalf. The donations are limited to 2gs, and the following amounts have been already promised Mr G. T, [sic] [Col. 1c–2a] Dunning, T<small>HE</small> S<small>PORTSMAN</small> Office, Fleet-street, who has been requested to act as honorary treasurer to the fund: [names set as unordered list to save space, last 3 column headers: £ ''s.'' ''d.''] * Lord Mandeville 2 2 0 * Lord Royston 2 2 0 * Hon. Michael Sandys 2 2 0 * Mr Fred Ketchie 2 2 0 * Mr Ike White 2 2 0 * Mr W. Riley 2 2 0 * Mr C. Parsons 2 2 0 * Mr J. Kilby 2 2 0 * Mr Frank Hinde 2 2 0 * Mr J. Calder 2 2 0 Any contributions sent to the hon. treasurer will be duly acknowledged in T<small>HE</small> S<small>PORTSMAN</small>, which in a couple of days will contain the names of a committee now in course of formation. In the London clubs yesterday the opinion was very general that Slavin ought to be compensated for the loss he had sustained in consequence of the ruffianly tactics pursued in Smith's corner. The utmost indignation was expressed in reference to the treatment the Australian had received at the hands of so-called Englishmen. We have received a number of letters on the subject. The following is a sample of them: To the Editor of T<small>HE</small> S<small>PORTSMAN</small> Sia, — I have no doubt that many Englishmen will have read with regret the disgracefully unfair treatment that Slavin met with in his encounter with the  "Champion of England" to-day. I trust that a subscription will be started to show him in a substantial manner that the sympathy of most Englishmen who take an interest in boxing is entirely with him. If you are prepared to receive subscriptions I shall have much pleasure in subscribing a guinea, and in collecting those of many friends who will be glad to subscribe. — I am, Sir, yours faithfully, C. B. C. Royal Thames Yacht Club, December 23. [We refer our correspondent to the paragraph above. — Ed. T<small>HE</small> S<small>PORTSMAN</small>. ] [sic bracketed interpolation] Eyewitnesses Interviewed. A representative of T<small>HE</small> S<small>PORTSMAN</small> was at Cannon-street station yesterday afternoon to meet the Continental express, in which a goodly number of those who had witnessed the fight made their return to town. The train was somewhat late, and did not steam into the station until a quarter to six, when our reporter singled out several well-known "sports" (including more then one scion of a noble house) for a few minutes' chat. Asked as to how affairs happened at the outset, one and all agreed that it was plain a row was meant, the demeanour of a certain section of the party, which  did not number more than forty, being particularly noticeable. "How did the combatants' seconds behave?" — "Well, after the first round, it was evident that Smith would soon be at the mercy of his powerful opponent, and his seconds very soon perceived this state of things. In the initial round Harper came down with some force on the Australian's chest, while in the next bout Slavin came in for some rough treatment at the hands of Baldock, the Champion evidently suffering with one of his eyes." "What form was the Englishman displaying?" — "From the outset he made a miserable stand, and was plainly afraid of his rival. He fought as a coward, and by the tactics adopted by several of those at the ring side Slavin was fighting against great odds.  Every time he got into Smith's corner he was met with remarks of a most uncalled-for character, and had to be very sharp to keep at arm's length of several rowdies, who, as the fight proceeded, became extremely noisy." "How do you account for such a noisy corner at the back of Smith?" — "Although they did not number more than eight, they were some of the worst  characters known, and had undoubtedly been commissioned to play the part they did. This was most disgraceful, isasmuch as they not only made Slavin a mark for the knuckledusters and other weapons they were armed with, but likewise made tracks for individual spectators who ventured to express their indignation at the course of proceedings. One rough, more venturesome than the others, "framed" up to a familiar figure at boxing tournaments, but on the latter producing a long bowie knife, accompanied with a few words of a somewhat emphatic nature, the villain rejoined his  party." Are these men known at all?" — "Yes, they are mostly from Birmingham, and are of the lowest order. It was reported that more than one had been kept at the country's expense, one for a very lengthy period." "Was not this sort of trouble expected?" — "It had been given out by one influential patron of the English Champion that there would be an absence of rowdyism, but it was also stated on good authority that certain roughs had been specially engaged, so that trouble was naturally anticipated." In reply to questions as to how the contest proceeded, it was the opinion that the thirteenth round would be the last, as Smith's party made a determined effort to bring about an undesirable finish. On the two fighters getting to close quarters more rough work ensued, and on some pretence Smith left the ring. Slavin, however, stood his ground manfully. "How did the crowd behave towards the finish?" — "There was a lot of excitement on all sides, but the only fear was that the rough element would resort to extreme measures in endeavouring to save their Champion. Others were ready to insult the Australian by using most filthy language towards him, and altogether affairs began to look serious." "Is it true that attempts were made to tamper with the water used by Slavin's seconds?" — "Yes! a man was noticed to be hovering about in that quarter, but, fortunately, his efforts to drug the fluid were of no avail." "Did Smith deal out much punishment?" — "His blows lacked the force of his rival, and when the referee gave his verdict as a draw the Australian looked very little the worse for his exertions, and could have fought far two hours more. He could have beaten Smith and both his seconds into the  bargain, given fair play." "Was Slavin injured during the fight?" — "No; not from the blows of his rival, as he heeded them very little. He, however, received a nasty blow from a knuckle-duster by one of the cowardly ruffians, which caused his ear to bleed." "As to the referee's decision?" — "That official was undoubtedly afraid of his life, and thought it best to stop the proceedings before any serious harm happened." "What becamiof Smith after the fight." — "He returned to his hotel, and will no doubt arrive by a later train. His friends were disgusted with the form he displayed, and it will be a long time before he recovers his former prestige, if ever he does." "How did the Cornstalk behave in the ring?" — "He fought in the most manly style throughout, and gained the highest opinion from all respectable people present. A testimonial to Slavin has already been decided upon, and no efforts will be spared by those concerned in making it worthy of the occasion." Mr Ike White, a well-known sportsman in the city, and one who, as a rule, does not mind going to considerable trouble to witness a "mill," was very severe in his comments on the gang who were so lost to all sense of decency and fair play. Mr White said: — "The story of the fight does not need much telling. Slavin had his man beaten at every point, and was winning easily. Slavin is a very brave man. He bad to go into Smith's corner every time, and when there had to fight not one but six men; the ruffianism was simply awful." "Do you think it was for the best that the contest was stopped as it was?" "Yes, certainly. Goodness knows what would have happened before long. Probably Slavin would have been killed, as the villains that had congregated in Smith's earner seemed determined, at all hazards, to prevent him from losing. More than once the Australian had Baldock on one side and Harper on the other, both jeering at him, whilst Smith was in front and the 'boys' were making all sorts of attempts to injure him, but he seemed to pay no attention to anyone but Smith. Slavin would not have had a mark from Smith, such as he does now having been inflicted entirely by the mob[.]" "What do you think of Slavin's abilities?" "He is a very great natural fighter, and as for his gameness a pluckier man never pulled off a shirt." On the subject of who was responsible for the presence of the rowdies that caused the bother Mr White was reticent, but not so when the subject of a testimonial to the Australian was mooted. "He is coming to town to-morrow," said the speaker, "and shall have a great reception. Some good sportsmen  have promised to leave no stone unturned to give Slavin a testimonial to be proud of, for he richly deserves it." "What seemed the general opinion on the ground?" "Why every ''man'' (with great emphasis on this word) was sick and disgusted: all but the mob gradually forsook Smith's side of the ring, and the Englishman will lose many friends in consequence of the turn affairs took." Reception of the News in Birmingham. Considering the ''penchant'' of the natives of Birmingham for fighting, and their interest in all matters concerning the ring, a stranger in the city yesterday knowing this would have been rather taken by sur prise [sic] at the indifference manifested in the result. At the various local sporting resorts, however, the cause was plain enough to those in the know. Early in the week it was an open secret that a warm corner was being provided for the English Champion, consequently betting was at almost a standstill. In what little speculation took place the odds were in favour of Smith, as the ''cognoscenti'' were confident that the Australian would not be allowed to win. Amongst the better class of sportsmen in the city the whole of yesterday's disgraceful proceeding were viewed with the greatest disgust. There are still many staunch supporters of the ring in the district who would gladly hail any signs of its regaining its past glory, but they look upon the fight of yesterday as the death-blow to the institution. By the few that thought the affair might possibly be allowed to be decided on its merits Slavin was expected to win, but his only chance was considered to be in his forcing the fighting at the start [Col. 2c–3a]  and knocking his man out early. George Prober hoped to hear that the fight had been decided fairly, and he regretted the unsatisfactory ending. How the Match Was Made On the evening of Saturday, August 10, the P. and O. steamship Carthage cast anchor in the Albert Dock, Limehouse. On bard was an individual, anxiously expected by the British public — namely, Frank P. Slavin, the Champion at Australia, who was accompanied by his manager, John Lewis. Without loss of time, Slavin made his way to T<small>HE</small> S<small>PORTSMAN</small> Office, and on the succeeding Monday the following interesting particulars respecting the visitor appeared in the columns of this paper:— "He has a long record, including victories with both gloves and knuckles, and has never yet known defeat. His bearing is that of a thorough gentleman. He is a fluent talker, and can converse on any sporting subject. Devoid of conceit, he yet has a full comprehension of his own powers, and speaks with confidence as to the result of any encounter in which he may take part. Slavin is a splendid representative of the nation which is growing up across the seas. The contour of his features is distinctly English, square cut and firm. Beneath an intellectual forehead are two deep-set piercing blue eyes, and his features are bronzed with the burning suns of the south. From this it will be gleaned that the Antipodean is a formidable-looking customer, especially when we add that he stands no less than 6ft 1in, and in condition fights about 13st 4lb. Long before his arrival 100''l.'' had been posted at T<small>HE</small> S<small>PORTSMAN</small> office for Slavin to box any man in England, but the money was waiting nearly four months before any of our big guns would cover it. Jem Smith, the Champion, turned a deaf ear to the straightforward offer of the Colonists, and Mitchell  was called away to Yankeeland to lend assistance to his friend Kikrain [?]. For some time it looked as though our visitor would have to return as he came, without giving anyone a hiding or receiving one himself, until a gallant little English middle-weight stepped into the breach to box for the honour of the old country, and agreed to face the stalwart Cornstalk. This was none other than William Goode, who in consequence of his politeness, urbanity, and elegence of dress was christened by his friends 'Chesterfield.' Goode had previously lowered the colours of another Australian, Tom [Less?], a much bigger man than himself, and, bearing this in mind, his friends freely plunked down 200''l.'' These were Messrs Robert Abel and Harry Hart, two well-known figures in the world of sport, articles after little delay being signed at T<small>HE</small> S<small>PORTSMAN</small> office, whereby Slavin and Goode agreed to box the bout of twelve rounds, under the usual rules, on October 17, for the sum stated. Many old ring-goers and patrons of the games shook their heads when the articles were published, and not a few blamed 'Chesterfield' for his rashness in tackling at catch-weights a man so much his superior in size. Goode, however, comes of a fighting stock, is full of pluck and stamina, and belongs to the order which never say die. Excepting in a competition, his maiden attempt, 'Chesterfield' had never known defeat, and the making a match with such overwhelming odds against him his bravery received due acknowledgment on all hands. Slavin held an unbroken record of nineteen successes, and throughout his career he had never known the effects of a knock-down blow. The articles stipulated that the men should box in a 24ft ring, and at the time of signing Mr B. J. [Angle?] was agreed to by both sides as referee. A very necessary clause was inserted, giving the referee power to award the stakes in case of what he might deem an organised interruption to the man who, in his opinion, had the best of the boxing at the time, and if all agreements contain such a clause, it will have a deterrent effect on those who invariably go on the sin, tie, and wrangle system. Slavin placed himself under the charge of that painstaking trainer and old-time boxer, Jack Start. The pair made Mr J. Kilby's, the Ferry Hotel, Cookham-on-Thames, their head-quarters, but things went very badly at the start. During the second week of his training the Australian caught a severe cold, and rheumatism setting in, the services of a doctor were required. The illness was, for obvious reasons, kept a profound secret, but so bad did Slavin get that the first man of medicine deemed it necessary to call in a second. Ten days were spent in the house before Slavin pulled round, and though all went well for some time, he contracted another bad cold. Start, however, wa unremitting in his attention to his charge, and Slavin came to the scratch in fairly good shape. The result of the encounter, a win for Slavin in the fifth round, despite a grand stand by Goode, will be fresh in the memory of our readers." [sic: quotation begins at the top of the paragraph] Smith's failure to make a match with the Australian caused general surprise, but the Englishman would have nothing to do with the glove business, despite the repeated offers of Slavin. The latter's friends, on the other hand, saw little but trouble attached to a knuckle fight, and though Slavin himself preferred the fists to the mittens, there appeared small chance of a match between the pair until Mr Frank Hinde stepped forward and put up a deposit of 50''l.'' at T<small>HE</small> S<small>PORTSMAN</small> office for the Australian to fight Smith for the championship and 200''l.'' a-side. The old question of stakeholder then cropped up, and, after a Iot of bickering, the Colonial gave way, and agreed that the money should be deposited with private gentleman, who for obvious reasons shall be nameless. A meeting took place, and eventually the following contract was drawn up by a representative of T<small>HE</small> S<small>PORTSMAN</small> and signed: Articles of Agreement entered into this day (Monday, September 16, 1889), between Frank Slavin, of Melbourne, Australia, and James Smith, of London, England. The said Frank Slavin and James Smith hereby agree to fight a fair stand-up fight according to the new rules of the London Prize Ring, by which the said Frank Slavin and the said James Smith hereby mutually agree to be bound. The said fight shall be for 200''l.'' a-side, open for 1000''l.'', lor the Championship of England and Australia, and shall take place in 1889 out of England. The two representatives to select the battle-ground issue all invitations (not exceeding fifty persons a-side), and carry out the arrangements for bringing off the fight. The backers of the said Frank Slavin and the said Jem Smith this day (Monday, September 16) deposit 200''l.'' with the stakeholder. The representative of each man to supply the other with a list containing names of spectators, and to have mutual power to strike out a name they object to. The said James Smith and the said Frank Slavin each to have power to appoint an umpire. The referee to be mutually agreed upon by the duly authorised umpires of the men at least seven days before the day fixed for the fight, but in case they cannot agree, the final stakeholder to have power to appoint a referee. The men to be in the ring between the hours of six a.m. and twelve noon, or the man absent to forfeit the battle money. The expense of the ropes and stakes shall be borne by the two contestants equally. In case of magisterial interference the referee, if appointed, or the stakeholder if not, shall (if possible on the same day) name the next time and place of meeting, and either party failing to appear at the time and place specified to lose the battle money. The stakes not to be given up unless by mutual consent, or until fairly lost by a fight, and due notice shall be given to both parties of the time and place of giving up the money. In the event of any question arising which may not be provided for in these articles the referee to have full power and authority to decide such question, his decision to be final, conclusive, and subject to no appeal in any court of law. The stakeholder shall in any and every case be exonerated from all responsibility upon obeying the direction of the referee. In pursuance of this agreement we hereunto subscribe our names. Frank Hinde (for Frank Slavin). John Fleming (for James Smith). Witness for Smith, F. Montague. Witness for Stavin, John Lewes. Subsequently Slavin, as stated, defeated Goode, following which he fulfilled an engagement at the Royal Aquarium extending over thirteen nights. The "Cornstalk," though suffering from an injury to his left hand, offered 20''l.'' each evening to any man whom he could not step in four thee-minute rounds, and though several tried to earn the money Slavin always accomplished his self-imposed task, and brought his engagement to a close in a blaze of triuumph by stopping two men the last evening in four rounds. Slavin had now come into funds. Through the columns of this paper he expressed a wish to put some of his own money down, and, confident in his ability to uphold the honour of Australia, he offered to increase the stakes to 500''l.'' a-side. Smith, with a millionaire at his back, had little difficulty in getting the extra 300''l.'', and accordingly the money was made 1000''l.'', the sum fought for by Sullivan and Mitchell. Smith in the meantime, though ignoring Slavin's challenge to box under Queensberry rules, strange to say, entered into an engagement with Jackson under the conditions he so stoutly objected to when put forward by the white man[,] His ignominious show with the coloured Champion may be passed over without comment, and it will suffice to say that after he and his conqueror had boxed together for ten nights at the Aquarium the Englishman began to prepare himself for the affair under notice. The Men in Training Slavin, when he entered upon his training at the Nayland Rock Hotel, Margate, was 12st 9lb in weight, while fourteen days later, in the buff, he scaled 13st 6lb, and was as hard as nails. The increased weight was accounted for by the splendid development of the muscles in the arms, shoulders, and back. His daily programme while at Margate was as follows:— Up at seven sharp, and twenty to twenty-five minutes' strong exercise with 4lb dumb-bells, followed by a gentle stroll of a couple of miles before breakfast, at 8.30, which consisted of porridge, 1lb of chops or steak, dry toast, and a cup of tea. After the perusal of the morning papers he indulges in a ten-mile walk at a smart pace, averaging five miles per hour. Back again to the hotel, into a bath, and then a rub down with whisky, which was completed in time to give him a good rest before dinner, timed for 1.30, and composed of either roast beef, mutton, or chicken, with vegetables, light pudding, and a pint of seven-year-old bitter. The meal being digested over a game of billiards, he started at three o'clock on a six-mile stroll, varied at intervals by sprints of one hundred yards. On his return to his quarters he punched the ball for half an hour, and then sparred lightly with an Australian gentleman who was at the time on a visit to this country, and holds the Amateur Championship of Australia, he having kindly consented to assist Slavin. This was followed by a sponge and rub down and a good slapping of the muscles before a tea, at 6.30 p.m„ [sic] of cold meat, half a dozen eggs, dry toast, and tea. The wind-up of the day was a quiet stroll to the pier and back, and a game of billiards before retiring to rest, at ten o'clock sharp. Smith's programme was something of the same description, excepting the ball punching, the Englishman preferring the sack. He made the Greyhound,  Newmarket his head-quarters, most of his exercise being taken on the Heath, though, as visitors were forbidden and interviewing tabooed, little was gained [Col. 3c–4a] as to his doings. Under the care of Jem Howes, his old trainer, he was, however, reported to be doing well, and much finer than when he fought Jackson. Finding the Battle-Ground. (By one of the Explorers.) At a meeting held on Saturday, November 16, Mr John Fleming and Mr G. T. Dunning were deputed to act on behalf of Smith and Slavin respectively, their duties being to select the battle-ground, issue invitations, and carry out the arrangements for bringing off the fight. This was rather a strong order, and one of a most trying nature, as anyone conversant with the peculiarities of those attached to the P.R. may know. Wednesday, December 4, was a cold, raw day, and with the thermometer down to freezing-point travelling at night was anything but a luxury. Reaching Paddington soon after 9 p.m., Mr Fleming and I, thanks to an energetic guard, soon found ourselves comfortable in a first-class "smoker" attached to the Weymouth express. At 9.15 to the tick we steamed away from the Great Western terminus, and in due time reached Reading. The hour had passed quickly, pleasantly whiled away by interesting tales of old-time bouts told by the manager of the Pelican boxing. Such men as Punch Dowsett and Bill Charlton graduated in Mr Fleming's school, and the way these celebrities used to set about each other for as many shillings as the boxer of to-day gets pounds is nothing less than remarkable. A ten minutes' halt at Swindon was utilised to the fullest extent, and then bowling along we reached the end of our journey, so far as land was concerned, soon after 2 a.m. on Thursday. When we left town the air was crisp and frosty, with hardly a breath of wind stirring. We were, therefore, greatly surprised when stepping on beard the Gazelle we were informed  that a "choppy sea" was running outside. The fact of a passenger's hat going over the side proved the truth of the assertion, and then this same passenger, a likely-looking fellow, bemoaned his fate and bad luck. "How confoundedly annoying,"  he said. "This is the sixth time I have been across during the year, and not once has it been calm." "Better have a sleeping berth, sir," said a sprightly youth at the top of the stairs. "Sleeping berth be hanged. I'll remain on deck," said he, and here for a time we left him inquiring if his horses had gone across in the morning. Guernsey, dear reader, was our destination, and with the idea of making ourselves comfortable we made for below. I felt a bit "peckish" and cast a longing eye on the good things set out in the dining saloon. This was, however, as far as I went in the line, and diving a floor lower doffed my boots and courted the god of sleep. Morpheus, however, scorned my attentions, thanks to the clatter on deck and the noise of the machinery, to say nothing of the "magging" [?] of our friend of the horses. This gentleman, after three or four futile attempts to drag me into a controversy about traveling, "struck ile [?]" with an elderly party, occupying the berth behind me, and for nearly an hour I had a pretty lively time of it. The "elderly party" had a voice greatly resembling the dividing of  tin hoops with a file, and in addition displayed a shocking habit of making horrid and atrocious puns. '''Just''' re 4 Vaught  al up sleep far led job aid  Ifni( sp.. the Gesell* is a meal  bread she seat the offsets of the keens  sad began to tel liesay. The aware in  the motion et the simmer was followed by a  deeded in the sessamilea mead as, and by  wise ben bar to wane. The  '4 4. •  tie slop dipped isle a walk. sad As  La boss amisaged tie two came to •  dad Meier up a payer of thenliegiving for  tide I eked my eyes sad bass a good Wier  was cid* *new the et, I lope, the jest.  el the  The eat mills "plod the Onerseey  Pier,  length  en adesigeeded by a prod frees Flem ingss did, at  ema -a  AMONG THE CHANNEL ISLANDS.  Tile thee by "Joe Vickers" was T aim, sad ludrawake  I took to my boots. 'Need .at boathook •we  slall be than is Imiautemaidesdi e  annoy,  sort el  • w gi l la ar Zlessedrnies ea ----- TWl — dsek. el  velar sad s cadre at b.l sad rag  mead se sod *ga mg g tate taw lise  = i n  I ea" i..iowt Mwes minto re fawn at Gardaer's  lam.  Raft vend a "shoals* (amide= knew what he  meet, Kenplp, wade a welcome, and •41balis e the  ammlsslowr be beaded vs over to or* of  M iamiesid. I tempt ha mem  eg ea here I awes* r oi l  I all esy ne awe Mrs thee after a cep of tea  with daises mess, we tsar a well-esened sad marsh  nestled ail. Up sal Ades -see after tin, a dilate  talk wee bellmeue. Iftenteg had a find WIWI and,  a erphia who lame' • spot when we add  • fee a sigma. Onessa, dear to the "honey  mane end &Ogallala, isomr." le. at wrtgaska _Ulm%  ihead lama alma le the IWO& Ckeend rime  ileasels Balking and Nr •to wl7. It hal Win  frees Weaneeth, Tftra lark 19 ban Ailersey, and  25 fresJaw. I assilen time dines/es, for we  =plead the oh= paw &ed with rata  alma had- in high =lse, ma  isw  thospeene tines nee the  th=4  el vide booeind err of an lime an  boa at Yr bard thelores loyal HOW as  Cestleasernet eras ia view, while hi  Ks di- ---s, sired Miles by • heavy waist, add  r k. he Mesesea Ks lima a Ham, Mien, a=  amid M sasatiorsi, though  ands the prison-- d Negkiod, bosses a sort of  Mesa Mae gpranagai, rallare frame to the  sierearer Ms pas, if that strasger is  es —°-ald bed and gets Makiza. ,, iii •  ands= aroser, bairns at the  heal • Ms= is " Ma* laid I, "He's • Frenaass."  "Gell eni." " hesa •berle.*"  "  had not e  Drinks beha  tins ash  al  hot  ," Dan ; aloft  bed  he is.ss " 3111*"  en.  to end bowl& a aillearresp sal a leoi.  la reply le • cal teem Flookag be opeeshed, and a  "Yea, an hens.," to I Arad hew to "park"  "Hike 1111/ • tamp, and I heed Mee from  the stark For elms Haan be had hen in  the serene of a es--el at 11. a week,  and tires snits of abase is the fir. Hotel pay  was ant so geed, sod Mash be wade "a hit" a the  somoner, the yaws an an generally there when  the bills an pad, if nee whoa the work was ea.  " Mike" told a hie el also to what seamy ewroldeo  the " foeolantinig brogue."' He lammed us—and  how ear barn hat at the news—that the Wand,  which had • pastaise of over 32,000, employed but  sir sPOMOINIMIC "There's not a begpr in the land,"  aid Mike, "and bat one Hasson on, and," antis/1i  to the window, "be ahem there it goes. " So  tars - bat shoe a walk round the eon of Brie  pp re i m e s: *et by saying, in reply to a query from  "They've a way of thdr own on this  and, sad mealy ivory ether men L .1  ee  3.4.11  nsertessoholy we bad Pat  nowt,table." W  and, •••••= ar easik ether, we  isele:a w  hilly to the doer, IMMO with  • breath of the pnes as  lasne w r i tZ  et  Mee steed ea ths base lie relhea  whilst imi 7wts  soaring free a "ladling" inn the heal of anything  bat a *ring =We. Videos et a dimgemes/  man Code Masa haled Moe*  a ales end a step were hand bsNsd, sod th I was  eneewhit resesored. "Mb= would yw like •  dolt sully on wise an Afraid tint we  inel ha-- *saga as bit week at the leig p r ammis ned to  I=her sad rid Inlet he swat.  11111 lewd tbs " ark " was • sort of Mak  ask ea wheels aid threighent haw did. Deinlag  the offerors leek • aid, AM path t  aid hart tegother, eine te the  that we tea was d a very doffs= name.  The lid dog was to the captain, who, in  reply te m : g a =Ma epee a. Repave  as et  tie the and = heleeisslien to  Kr H., a who w wellertenally lowed =my  Omi  front holes. Tang a " ask "we thee sleeted en a  voyage of apieraties, V  the  eve way to  • sae nsam. Ine lit. Fenn het aid  ran lireigh Bt. Bona= s—the ffebs—ree iae  Germs ewes If Awards, in one buboes  ale shag the readahle. MI  : 6 =4  drive ▪ y ve be a es& les to the onnes• assilmol, end Mr  = we Weed van to a lewd plead tat.  sheraied on al ales, and pleetto en we  ameba liAnnues Bay, • body Kite spat, _ when  the sea rippled softly over a bed of almost mew-white  mad. Near= could be fennel a suitable sea where's  t i V "1"4/P le e ," and es ibe ANON  al were f lest we rattled hack  te the "D asp A alga offered no imbi•••  en. for further travel, and it was a an  of early to bed and early to rise, the next  Fleming end arson at the door of Mr H.  *Obi goodkowa wall the Nature of  the ale b  usliun al  hand be, Eke a owed Ipseleown, fell into the  and at own vohniessed to drive an to " the best  in the world." "Dodo" was dummied to the Alb.  cart, and away we went at the rate of fully fearless  miles an hoar in the direelissof the Forest. I have at  behind many a pea goer, bet must for her size award  the palm to Bess* a bea uty freely twelve bandahigh.  Throwingthe robe in the helium of the vehicle, Mr IL  let the pony have her bead, and the way she went  uphill, down dale, and past other traps was truly  astonishing. Great Bt. Peter's was passed, and lovely  scenery encountered, the gardens being full of flowers,  ceraniums and roses blooming on all sides. An enjoyable  drive of forty-five minutes found us at Boe(pair.  Bay, and directly in frost stood the island of  Lihou. As many of our readers may know, Lihou is a  small hill of land that at low water can be reached  on foot from Guernsey. As a fact, a paved roadway  ' is laid in the bed of the ocean, across which both  foot passengers and carts easily make their way.  With the tide up communication is effected by boats,  but this being low at the time we had the novel experience  of  DRIVING THROUGH THE REA.  Gaining the other side we " explored " and as the  mew of land is only some three hundred yards by •  hundred and fifty, we were not loag shoot it. Two  excellent patches of level turf were found, but just aa  we were making for the trap • punt, Robinson Cruse&  looking individual scaled one of the small stone walls  and barred the way. Of course, he wanted to know  our Wainer, and, after • rather rough opening,  quickl " listened to reason." In the course of wovrrtian  ws found that the population of the Wand  was limited to three, the old "jam,. " and his two  daeghtere, with a " moke". thrown in. Pointing a  long skinny finger , which might have belonged to the  Witch of Endat, towards the quadruped mentioned,  the old man said, " Us two keeps the family.  He draws the seaweed and I born it."  "AM why do you been it ? " " Welk to looks  air, end base be ashes adios Y amaefactured."  " And hew wadi de yen are • weak by that 7" said  I. "Oh. .be Sr. ; not fw. e y weak." Great  Scott ! and MB was all the old man, his tat mace-,  sad the " mite" had to live on. bi, •  fleas sash. wash needy aupelied the old  PleZi y end =MN sensed to Kr M., who  es lea to AX Pate's Part. The next day we  messed le Sark and Jersey, hut fazed nothing no  =Seib es the panel at and fairly well seta  bed with the resell of ene abet, • day later boarded  the Lyle: hr the sawn bemey.  = air visit the mat= had tea of a fairly  fismelplisa ha the voyage home was made  sear very deprening drenmsteame. Rain fell heavily  sod slack mist brag ear lite water, which after rag  a few =los was saw alb porpoise. Some of these  animals ease within • few yards of the Lynx, but the  speed of be boa preyed too pod for the linney saes, I I  who were penury left behind. With • bass  springing up sal was hated, and dashing through • I  heavy ma to spi=did fern the Lyon made some sixteen  keels be the her. Ia this goy bait the journey 1  was semplead, when all d • sadden we were caught  in a anal. In as Mint the and chopped round,  and before you amid "Jack Robins= " the good  vessel was tamed arweiii to. The order to take in  sail was promptly obeyed, bat the clatter on deck  brought may an IMZIONI passenger rapidly from  below. A very heavy sea was now running, and  though the Lynx kept her course bravely the decks  were swept from stem to stem. A huge Eat Indiaman  now hove a fugh• t. and. lie one at the men  informed me, she had the audacity to cress  our bows instead of passing ender our stern. This  breach of ocean =goats =Sod the skipper of the  Lynx, who chased the trader, and eventuall y made  her alter her coarse. This little "diversion "caused  much amusement amongst our passage= and after •  cold, rough passage we laaad safely at Weymouth.  Chilled to the bone, we made for the Victoria Hew,  when a chop and a glees of old October revived the  almost Mamma circulation. Loudon was reached  about 9.30, aad, tired and worn out, I was glad to lay  my bead on • pillow I had not sego for four days. === Dundee ''Evening Telegraph'' === This Wednesday, 25 December 1889 article describes Slavin's reception in London after the fight. Viscount Mandeville was there, suggesting that he was a Slavin supporter:<blockquote>There was (says a sporting correspondent) great excitement in London yesterday over the fight of Monday, and all classes of the community expressed their abhorrence of the proceedings adopted by the Smith party. It was known early that Slavin would journey up from his quarters at Margate about mid-day, and by two o'clock there was a large gathering in and about the Holborn Viaduct Station. By three o'clock the crowd had increased enormously. The train arrived behind time. As soon as Slavin and Mr Taylor were recognised, a loud cheer was raised. So great was the crush that it was only with the greatest difficulty that the Australians could make their way to the street. The reception accorded the champion, although very gratifying, was somewhat rough, Slavin and his party being forced across the road before they could take refuge in a hansom. Among those present were John Lewis, Slavin's manager; [[Social Victorians/People/Mandeville|Lord Mandeville]], E. Bayley, and other well-known sportsmen. From the station Slavin went, in answer to an urgent request, to the Stock Exchange, where another enthusiastic reception waited him. A move was then made to the office of the ''Sportsman'' in Fleet Street. Among those who were present to offer their sympathy and congratulations to Slavin on what all considered to be a victory were [[Social Victorians/People/Mandeville|Lord Mandeville]], Ike White, E. Bayley, George Knight, W. Riley, C. Mitchell, J. Goldberg, J. Kilby, F. Symes, F. Hinde, and a score of well-known boxers. Shouts for Slavin were kept up outside, and the crowd could not be silenced until Slavin went to the window. He bowed his acknowledgments, and after much difficulty obtained a hearing. He thanked the multitude for his great reception, at which three cheers wore raised for him, and a corresponding number of groans for Smith. Later Slavin dined at the Albert Club where a Testimonial Committee was formed. At seven o'clock Slavin departed for Margate. Smith remained in Bruges until one o'clock on Monday, when, in company with Baldock and Harper he left for London ''via'' Calais. They arrived in London at eleven o'clock, Smith going home in company with a friend. Smith has a wound over the right eye, and his face is much swollen.<ref>"London Gives the Colonist a Great Reception." ''Dundee Evening Telegraph'' 25 December 1889, Wednesday: 3 [of 4], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18891225&page=3&article=052&stringtohighlight=mandeville+viscount. Print title: ''TheEvening Telegraph'', n.p.</ref></blockquote> === ''Reynolds's Newspaper'' === The Sunday, 29 December 1899, of ''Reynolds's Newspaper'' published a detailed account of the fight.<blockquote>The fight between Slavin and Smith took place on Monday morning upon private property in the outskirts of Bruges. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, had placed his grounds at the disposal of the combatants. The host is a well-known breeder of fowls; and he certainly did not expect the quality of fighting cock which responded to his invitation. The ground was encircled by large trees, which secured the utmost privacy. About forty persons assembled at the ring-side, including Viscount Royston, Viscount Mandeville, Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. P. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known "sports." Smith was dressed first, and had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots; while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vesey officiated as referee and Mr. Calder as time-keeper. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently and surprised many of the spectators by the determined way in which he fought. In the thirteenth round he doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of "Police!" Smith was hurried from the ring; but Slavin and his friends refused to budge, and, after "Time" had been called, they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. Prior to this Slavin had been repeatedly kicked on the legs and hit on the body and head with sticks and knuckle-dusters. These gross and cowardly outrages were committed in Smith's corner, where all the fighting took place from start to finish. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing around beat and kicked him in a most cowardly fashion. Amid cries of foul, the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the light a draw, and, crossing to Slavin, said, "I do not want to stay here to see you killed." Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to Margate. Smith and his friends were left behind at Bruges. Round 1. — Slavin stood over Smith with a longer reach. Both sparred cantiously at the opening, Slavin at length leading off and landing heavily on Smith's chest, jumping back out of danger. Sparring again Slavin landed full on Smith's mouth. Heavy in-fighting followed, the men closing and falling together. When Slavin rose it was seen that he was bleeding from the mouth. Time, 1min. 30sec. Round 2. — Slavin fought Smith right into his own corner. There was a disgraceful scrimmage there among Smith's friends, Slavin being badly used. Both men fell together. Time, 15sec. Smith's party shouted out that their man had been spiked by Slavin's boots. The referee, thus appealed to, declared that the men had spiked each other. Round 3. — Smith showed to better advantage in this round, getting home both right and left twice in quick succession on the Australian's body. Sharp close in-fighting followed, Smith going down before a terrific right-hander. (First knock-down blow to Slavin.) Round 4. — Both men were blowing when they came up for this round. The fighting was again fast and furious, Smith at length cross-buttocking Slavin. The Englishman, however, did not fall on him. Round 5. — A light round, Smith going down before a light blow. Time, 11sec. Round 6. — Slavin came up very determined, and forced the fighting at a great pace. He drove Smith all over the ring, landing with his right a heavy blow over the right eye. Smith went down bleeding freely from a gash on the forehead. Round 7. — Smith came up stronger and did some of the forcing work, beating Slavin for a moment on to the ropes. While in this position one of the crowd of roughs deliberately kicked the Australian, who, however, took no notice of the cowardly attack. Both the combatants got away from the ropes and fought to the middle of the ring, where Smith slipped, and both were taken to their corners. Round 8. — Slavin did the opening work in this round, driving Smith round the ring and finally into his corner, where, as usual, his friends were ready to protect him. Smith came out again, when Slavin landed heavily with right and left, and Smith sought the ground. Round — 9 There were heavy exchanges from the outset, Slavin giving his man no rest, and driving him back into his corner. While there one of the mob struck Slavin on the head a violent blow. Time, 1min. Round 10. — This was a far more sportsmanlike round. Smith stood up well in the centre of the ring, and fought it out without the assistance of his friends. There was hard fighting from first to last of a close order, both getting home some terrific body blows. Both fell together. Round 11. — Smith looked tired and led off feebly while Slavin returned with right and left on the head, cutting open the Englishman's cheek on the left side. Smith landed his left again, Smith countering heavily. Smith then rushed upon the Australian, who met him with a severe blow on the mouth, and then sent him down with a terrific right-hander. Round 12. — The disturbances at the ring had now become so unbearable that the referee cautioned the seconds to keep to their corners, and not to interfere with the principals during the progress of the round. The round was of the cautious order, Smith finally being sent to grass again from a heavy right-hand blow on the jaw. Time, 1min. 30sec. Round 13. — Slavin again did all the leading, fighting the Englishman round the ring, and forcing him into his corner. There were some heavy exehanges, Smith's friends again assisting their man by hitting freely at Slavin. Round 14 and last. — Just a second before "Time" was called, Slavin, remarking on the treatment he had received, said, "Are you Englishmen, boys?" This was received with derisive shouts from Smith's party, and then, in an instant, a cry of "Police" was raised. The whole of the Smith party instantly decamped, hurrying their man along with them. Slavin, however, declined to stir, and his friends shouted that it was only a ruse to stop the fight. After a time the Smith  party returned, and Slavin then said to Baldock, the most prominent of Jem Smith's seconds, "Why don't you give me fair play?" Slavin's seconds claimed the fight, but the referee ordered them to fight on. The tactics of the Englishman's supporters were now very evident, but Slavin, who was only anxious to have the fight out to an end, advanced into the  centre of the ring, and again beat the Englishman back into his own corner. A scrimmage ensued, both men landing heavily, and the mob in Smith's corner then closed upon Slavin, who was again kicked and struck. The Australian, however, paid no attention to anyone except Smith, and he finished up the round by throwing him heavily. The referee then said, "I shall refuse to act any longer. This man" — pointing  to Slavin — "can't have fair play. I declare it a draw." Smith seemed well at the finish, and Slavin was certainly full of fight and disgusted at having been robbed of victory by the cowardly attacks of roughs. The Australian had fought in the fairest and most plucky manner throughout, forcing the Englishman back into his corner in nearly every round, and keeping him there. Here the fight came to an end, the referee exclaiming, "I shall refuse to act any longer. I shall declare it a draw." There was considerable surprise among Slavin and his supporters, and the Australian said, "I came 16,000 miles to win, and why don't you give a man a chance?" After some little delay the company dispersed. The time at the end of the fight was eight minutes to nine. Mr. Cecil Raleigh, hon. secretary of the Pelican Club, 21, Denman-street W., writes as follows:— "As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it." Directly the news became known in London the greatest indignation prevailed in all the clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognizance of the arrangements for the fight, and also for the summary punishment of those who had dragged prize fighting down to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the committee. That Slavin has now a right to the title (empty though it be) of "champion" is clearly supported by the 24th rule of the Prize Ring, which runs as follows:— "Any pugilist voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight." It seemed a pity that Mr. Vesey should not have had sufficient support to enforce what was uadoubtedly the judgment of all impartial spectators of round thirteen. Arrival of Slavin at Margate. Slavin arrived at Margate at half-past seven on Monday evening, and met with a flattering reception. During his training in the town he had become a general favourite with those who took an interest in the event. Sympathy was almost wholly on his side, and during the day news was anxiously awaited. At the station he was met by a four-in-hand and the town band, and proceeded to the Elephant Hotel, the band playing, "See the conquering hero comes." Along the Parade, on which was assembled a large crowd of people, he was loudly cheered. At the Elephant Hotel, there was a large crowd of people, and the reception was most enthusiastic. Slavin briefly addressed the gathering, thanking them for the reception he had received. His intention, he said, when [Col. 4c–5a] he left England was to win the championship or lose it fairly, and he was sorry that the fight had ended in such a fiasco. Inside the hotel the scene was very exciting, and Slavin received the hearty congratulations of his friends at the slight punishment he had received. He then left for the Nayland Rock Hotel, his quarters, and on the road was loudly cheered. Upon arriving at the hotel, Slavin retired, accompanied by his second, Joss Cosnett, after which the correspondent of the Press Association was introduced to him. The first question put was, "Well, Slavin, how have you fared?" "Very badly," responded the Australian. "As far as the roughs were concerned, I never witnessed anything more disgraceful in all my experience. My treatment at their hands has been most cruel, and nothing I could say would sufficiently emphasize what I mean. I am very pleased indeed at  the reception I received on my arrival at Margate. Nothing could have been more enthusiastic. You see what punishment I have received. It is not from Smith so much as it is from the outsiders." — Joss Cosnett: Smith did not bit him any more than I am hitting you. The conduct of the roughs round the ring was shameful. Slavin: The treatment I have received from them was most unmanly. I am sure the roughs never meant it to be fought out. They intended to have a draw if they found their man losing. Several of Smith's backers, including two well-known noblemen, were disgusted, and threw his colours away and put mine on, as a protest, I suppose, against the treatment I had received. To speak the truth, I would not like to see a dog treated as I was. All I am anxious for now is to meet Sullivan for the championship of the world for £500 to £1,000, or anyone else. I am willing and ready to meet and fight anybody for the championship before I go back to Australia. The Pelican Club and the Fight. A meeting was held at the Pelican Club at one o'clock on Tuesday afternoon. At the conclusion the hon. secretary, Mr. Cecil Raleigh, furnished the following report:— "In view of the reports of what had occurred at the Smith and Slavin fight a special meeting of the committee of the Pelican Club was instantly convened by telegraph for this afternoon. There was a large attendance of the committee, among those present being the Marquis of Queensbury, Mr. John Corlett, and other well-known sportsmen. It was unanimously determined to address a letter of sympathy to Frank Slavin, expressing the appreciation of the committee of the courage he exhibited, in face of the most  blackguardly conduct which ever disgraced the ring side. The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck. The committee then determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as master of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the club on Tuesday, January 6th. The indignation of the members of the club knows no bounds, and every effort will be made to discover who were the real instigators of the disgraceful business." It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club move into their new premises. The following letter has been addressed by the pugilist, F. P. Slavin, to the Pelican Club:— "In reply to your very kind and encouraging letter, accept my warmest thanks for the high compliment you are about to pay me in presenting me with a belt, called the Champion of England belt. You may rely it will be my duty, as it has been all through my career, to do my best to win any contest I may enter into in a fair and manly manner. — Again thanking you, I am, sir, yours truly, "F. P. SLAVIN. "Nayland Rock Hotel, Margate." Mr. Fleming's Statement. Mr. Fleming states that the fight was arranged before the boxing match with Jackson. He went over to Belgium and selected the ground. As regards the rough element present, he did not know one of the men. No doubt they were engaged by some friends of Smith. The only persons who went across with them were four or five gentlemen. All connection between Smith and him had ceased, as he felt that he could not act in any way for Smith after what had occurred. Every fight that he had previously had the management of had always been brought off without the intrusion of any of the rough element at the ring side. He took no part in the proceedings, being merely there as a spectator, and he had not seen Smith since the fight. He did not think that he should have any difficulty in proving that he had no connection with the disgraceful proceedings at the ring side. The Preliminary Arrangements. The preliminaries of the match will, perhaps, be interesting. When, in 1885, Jem Smith, who had made a name among English heavy-weight boxers, met and defeated Jack Davis, of Birmingham, there was a sign of a revival of pugilism in this country. This was increased when Smith fought Alf Greenfield, and, although the battle ended in a draw, it was not long before the "lost art" was freely discussed, and there was every appearance of the revival of the prize-ring. Smith, as will be well remembered, fought a memorable battle with Jake Kilrain, who came from America to wrest the championship honour from the Englishman. John L. Sullivan, the Boston Boy, soon followed, and, although he failed to beat Charles Mitchell, his claim to the title of champion of the world was made good last July, when be met and defeated Kilrain. Then came new stars in the firmament of boxers, but glove-fighting had almost extinguished the "old style." Prominent among the late arrivals were Peter Jackson, a coloured man, and Frank P. Slavin, both hailing from Australia. Their records were of the highest calibre, Slavin having been successful in all the nineteen contests in which he had engaged, either with gloves or with bare hands, while Jackson's was but little less noticeable. The latter, however, carried all before him in America, and then sought fresh fields in England. Only a few weeks ago he vanquished Jem Smith in a couple of rounds. Slavin arrived in London a week prior to Jackson, and, after a lengthy wait, he was accommodated with a match. No one but W. Goode cared to meet him with the gloves, and Goode was easily disposed of. After signing articles to fight Jem Smith under prize ring rules, both men went into training, Smith at Newmarket, under the care of Jem Howes, and Slavin at Margate, with Fred Symes as mentor. The most difficult part of the arrangements had yet to be settled, and it was not without great trouble that a battle ground could be selected which afforded facilities for the accommodation of a party of sportsmen to witness the contest. At length a point was pitched upon near the Belgian capital, easy of access and fairly safe from interference. Another difficulty now arose over the appointment of a referee, and no less than five good men were named, all of whom were in turn objected to by Smith. The possibility of this having been foreseen it had been stated in the articles that the two parties failing to agree, the stakeholder, Mr. C. W. Blacklock, should appoint that official. This he did with satisfaction to both parties, naming Mr. J. Vesey, a well-known sporting man, to act. The stakes had then been increased to £500 a side. There is a great difference in the physique of the two men. Slavin stands 6ft. 1in., and scales in condition thirteen stone, is a very muscular athlete, and particularly quick on his legs; while Smith is of the sturdy order, and is only 5ft. 8¹/₂in. high. His weight when fit is 12st. 10lbs., and although his muscles are very massively developed, giving proof of great strength, he has not the quick hitting power of the Australian. Performances of the Men. Frank P. Slavin, the champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft. 1in. in his stockings, and in condition scales 13st. 5lb. or 13st. 6lb. Shavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. After winning several competitions as an amateur, Slavin's first fight was with knuckles against M. Power for £50 a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with two-ounce gloves for £200 a-side and the championship of Queensland, winning in three rounds. This was in December, 1885, and in 1886 he went to Sydney and entered into an engagement with Larry Foley. Here Slavin met Taylor, a pupil of Jackson's, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big follow named Bligh for £10 a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for £100 a-side, and Slavin won in nine rounds. He next met Mike Costello, an American, a good man, and after a big battle in Sydney, lasting two hours and a half, a draw was the result. Another match was made for £400, and £50 a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for £25 a-side to stop Costello in six rounds, but the latter is very clever and tricky, and lasted the time. Subsequently Slavin journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for £100 a-side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for £250 a-side and the championship. Jackson's party wanted a match with gloves for a purse. They could not, however, come to terms. In January last year the Australian made a trip to New Zealand. The first man put up was Smith, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the champion. The pair were matched for  £200 a-side, and fought with two-ounce gloves at Wanganui, Laing failing to come up at the end of the fifth round. After this Slavin returned to Melbourne, and met Jack Burke, but after Slavin had had all the best of it, the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for £200 with small gloves, which ended in Mike being knocked out in 2min. 40sec. Slavin returned to Melbourne, and fought Burke, Slavin undertaking to stop him in six rounds; £200 a-side was put up and the two fought with six-ounce gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for £200, with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. He then received forfeit from Woolf Bendoff. An easy victory over Bill Buck in two rounds followed. He then came to England, and his late victory in his glove fight with the plucky young Englishman, W. Goode, must still be fresh in the memory of our readers. Jem Smith was born in Whitecross-street on January 21, 1863. He stands 5ft. 8¹/₂in. high, and weighs in condition 12st. 10lb. Round the chest he measures 43¹/₂in.; waist, 36¹/₂in.; hips, 41in.; thigh, 24¹/₂in.; calf, 16³/₄in.; biceps, 15¹/₂in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing after one or two rough-and-tumbles with his fellow-workmen, he was taken in hand by Jack Knifton ("eighty-one tonner") and later by Mr. F. Grimm, of the Central Club. His first success was in a competition at the Griffin, Shoreditch. Later on he beat Snowey, of Holloway, and was then matched to box Harry Arnold with gloves for a £10 purse. The affair was looked upon as a good thing [Col. 5c–6a] for Smith, but it took him nearly an hour to polish off this last opponent, though he had slightly the best of the encounter from start to finish. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for £25 a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end of twelve  rounds. Smith also won Tom Symonds's All-comers' Competition on February 26, 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. Smith's next battle was with Jack Davis, for £100 a side and the championship, on the turf, and came off on the borders of Surrey and Sussex on Wednesday, December 17. In all six rounds were fought, occupying 7min. 56sec., Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for £300, at Maisons-Laffite, near Paris, on Tuesday, February 16, 1886. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Knifton, the "eighty-one-tonner" as he is called. This proved a fiasco. The memorable struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19th, 1887. This brings us to his ten-round glove contest with Jack Wannop, Smith laying £250 to £200, and winning easily. After which, as all the pugilistic world knows, Smith was easily defeated by the coloured champion, Peter Jackson, in a glove fight at the Pelican Club, the last named winning on a foul by Smith, who was at the same time well beaten.<ref>"The Great Prize Fight. £1,000 in Stakes. Brutal Scenes in the Ring." ''Reynolds's Newspaper'' 29 December 1889, Sunday: 1 [of 8], Col. 4a–6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18891229&page=1&article=006&stringtohighlight=mandeville+viscount. Same print title, n.p.</ref> </blockquote> === The ''People'' === On Sunday, 29 December 1899, the ''People''<blockquote>Disgraceful Scenes. Expressions of indignation. The prize fight between Jem Smith, who poses as the fistic "champion of England," and Frank Slavin, the Australian champion, took place early on Monday, and resulted in perhaps the most disgraceful scene ever witnessed in connection with the history of the prize ring. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, Belgium, had placed his grounds at the disposal of the combatants. In London it soon became known that the principal supporters of the Englishman had caused a number of Birmingham ruffians to be hired for the purpose of making a "strong corner." The betting changed directly to short odds on the Australian, and also evens, while  offers of 30 to 20 that "Smith did not lose" proclaimed that some ugly and disreputable manœuvres would be attempted. The ring was pitched on a splendid piece of turf — the lawn tennis ground, in fact — and about forty persons assembled, including Viscount Royston, [[Social Victorians/People/Mandeville|Viscount Mandeville]], Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. F. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr. E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known "sports." Smith was dressed first, and and [sic] had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots, while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle-holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vezey officiated as referee and Mr. Calder as timekeeper. The Fight. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently, and surprised many of the spectators by the determined way in which he fought. Smith looked very serious on coming up for the fourth round. At this time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. Getting to close quarters, Slavin delivered heavily right and left on Smith's body, the latter being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell. In the fifth and sixth rounds Slavin again brought Smith to the ground, and made a nasty gash on his forehead. All this time the most horrible language proceeded from Smith's corner. In the seventh round a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, knuckle-dusters and life preservers being used. Slavin during the progress of the eighth round further received a couple of smart blows from a stick which a cowardly ruffian carried. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's seconds following him about and jeering in his face. The referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith was one more brought to the ground. In the ninth round the Australian again ventured into Smith's corner, and the fighting brought him close to the ropes. The  Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely, bringing the Englishman to grass. The tenth and eleventh rounds were not quite so exciting, but in the twelfth Slavin drove his right with terrific force on to Smith's chin, knocking him clean off his legs. In the thirteenth round the Australian doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of " Police!" Smith was hurried from the ring, but Slavin and his friends refused to budge, and after "Time" had been called they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing round beat and kicked him in a most cowardly fashion. Amid cries of "Foul," the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the fight a draw, and, crossing to Slavin, said, "I do not want to stay here to see you killed." Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to  Margate. Slavin's Opinion of his Treatment. The following conversation took place during an interview with Slavin, a few hours after the fight:— Did you expect any trouble at the ring  side? — Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. — Do you feel any effects of the fighting? — Slavin: The fighting? I feel nothing of what I got from any one in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. — Why did you persist in going to his corner? — Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. — Of course you heard a mob was got together for Smith? — Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. — What do you think of Smith? — Well, I think so much of him that I'll bet him £200 I stop him in six rounds, but we must have no more private shows. I've had enough of that business. — l suppose you have a poor opinion of English fair play? — Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. — What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. — With the gloves? — No. I want to fight him with the knuckles. I think I should get fair play out there. Indignation in London. Directly the news became known in London the greatest indignation prevailed in all the West-end, [Col. 4c–5a] clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognisance of the arrangements for the fight, and also for the summary punishment of those who had dragged dowr [sic] a manly pastime to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the cornmittee. The Pelican Club Meeting. At the special meeting of the committee of the Pelican Club it was unanimously determined to address a letter of sympathy to Slavin, expressing the appreciation of the committee of "the courage he exhibited in face of the most blackguardly conduct which ever disgraced the ring side.["] The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck[.] The committee determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as manager of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the Pelican Club on Tuesday, January 7th. It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club will move into their new premises. While Salvin was at the Stock Exchange on Tuesday, a subscription in his interest was started, and £110 was quickly subscribed. Slavin Challenges J. L. Sullivan. Frank Slavin, the champion of Australia, is prepared to meet John L. Sullivan, under London prize ring rules, for £500 or £1,000 a side and the championship of the world. Slavin has lodged a sum of £100, and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr. R. K. Fox, of New York, and the money lodged with him or the editor of the ''Sportsman'' will lead to a match.<ref>"The Smith-Slavin Fight." ''The People'' 29 December 1889, Sunday: 16 [of 16], Col. 4a–5a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000729%2F18891229&page=16&article=150&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> === The ''Sportsman'' === On Monday, 30 December 1889, the ''Sportsman'' printed a list of people and organizations that had contributed to the "testimonial" fund for Slavin as well as some letters to the editor about the Slavin-Smith match:<blockquote> A committee meeting of the above was held on Saturday afternoon, Mr W. J. Innes in the chair.  There were also present Mr W. Riley, Mr Fred H. [Gala?], Mr C. Parsons, Mr A. Butler, Mr M. J. Harrow (Margate), Mr Ike White, Mr E. C. Irish, and Mr G. T. Dunning (hon. sec. and treasurer). The chairman read a letter from Mr John Lees, regretting his inability to attend owing to illness, whilst the remaining committeeman, Mr E. W. Bayly, was absent in Paris. The appointment of the hon. sec. and treasurer was confirmed by the committee, and after some discussion it was decided to close the subscription-list on Monday, January 20, and to present the proceeds to Slavin at the Cambridge Music Hall, Bishopsgate-street, on a day to be named. Other business of minor importance was transacted, and after the next committee meeting had been fixed for Wednesday, January 15, the meeting closed with a vote of thanks to the chairman. Additional subscriptions to the fund, which now reaches 300''l''., will be found appended [names set as unordered list to save space, £ ''s.'' ''d.''] * Lord Mandeville 2 2 0 * Lord Royston 2 2 0 * Lord James Douglas 2 2 0 * Hon. Michael Sandys 2 2 0 * Fred Kotchie 2 2 0 * Mr Ike White 2 2 0 * Mr W. Riley 2 2 0 * Mr C. Parsons 2 2 0 * Mr J. Kilby 2 2 0 * Mr Frank Hinde 2 2 0 * Proprietors of The Sportsman 2 2 0 * Mr. J. Purdey 2 2 0 * Mr. E. W. Bayly 2 2 0 * Messrs H. A. Domby and Son 2 2 0 * "Caption Coe" 2 2 0 * Mr C. W. Bulpett 2 2 0 * Mr P. W. Carr 2 2 0 * Mr W. Morley Pegge 2 2 0 * W. D. S. 2 2 0 * J. E. S. 2 2 0 * Mr W. H. Grenfel 2 2 0 * Mr Quartermaine East 2 2 0 * Mr H. H. Hungerford 2 2 0 * Mr Noel Fenwick 2 2 0 * Sir John Leslie, Bart. 2 2 0 * Mr T. Brinkman 2 0 0 * Mr Calder 1 1 0 * Mr H. W. Hill 1 1 0 * Mr John Young 1 1 0 * Mr D. A. Neilson 1 1 0 * Mr W. Neilson 1 1 0 * Mr Julius Simon 1 1 0 * Mr J. Edge 1 1 0 * Mr H. Marshall 1 1 0 * Mr F. Aynsley 1 1 0 * Col. Hon. Wellesley 1 1 0 * Mr Charles Hawtrey 1 1 0 * Mr W. F. Ashton 1 1 0 * Mr Barratt 1 1 0 * Old Durham Ox 1 0 0 * Sir John Willoughby 1 0 0 * A friend, per C. H. A. 0 10 6 * Mr Ben Giles 0 10 6 * Mr Tom Vale 0 10 6 * [E.?] W. 0 10 6 * Officer 0 10 0 * Per Mr Ike White, Exchange Club: ** S. W. 2 2 0 ** Mr G. B. Knight 1 1 0 ** Mr A. Powles 1 1 0 ** Mr S. Blandford 1 1 0 ** T. F. W. C. 1 1 0 ** Mr Leo Harward 1 1 0 ** Mr J. Harward 1 1 0 ** Mr H. Harward 1 1 0 ** Mr [Monteagle?] [Moutcagie??] 1 1 0 ** Mr C. Orate 0 10 6 ** W. G. P. 0 10 6 ** H. R. T 0 10 6 ** Mr Josiah Smith 0 10 6 ** W. H. R. 0 10 6 ** Mr R. Wilson 0 5 0 * Per Mr M. J. Harlow, Margate: ** Mr A. H. [Cosi? Coni?] 2 2 0 ** Mr J. Thurston 1 1 0 ** Mr J. A. Fagg 1 1 0 ** Mr F. [Miles?] 1 1 0 ** Mr Adolphe [Drineqbeir?] 1 1 0 ** Mr Ernest [Drineqbier?] 1 1 0 ** Mr Lacy Brown 1 1 0 ** Mr Algy Brown 1 1 0 ** Mr M. J. Harlow 1 1 0 ** An Admirer 1 1 0 ** Mr J. W. Robinson 1 1 0 ** Mr W. Searles 1 1 0 ** Mr J. Smith 1 1 0 ** Mr J. Hume 1 1 0 ** Mr J. Painter 1 1 0 ** A Scotchman 1 1 0 ** Mr J. Mundell 0 10 6 ** Mr Bob Redman 0 10 6 ** Mr G. A. Body 0 10 6 ** Mr J. Phipps 0 10 6 ** Mr J. Doughty 0 10 6 ** Old Fisherman 0 10 6 ** Mr Joe Knott 0 10 6 * Per Mr B. J. Angle ** Some mebers of the Stock Exchange 125 0 0 ** Hr Henry J. Turner 2 2 0 * Per Mr W. J. Innes: ** Mr Sidney Fry 2 2 0 ** Mr W. J. I. 2 2 0 ** Mr W. Schlesinger 2 2 0 ** Mr Joseph Thompson 2 2 0 ** Mr Barney Thompson 2 2 0 ** Mr L. Meyers 2 2 0 ** Mr W. Shoe 2 2 0 ** Mr George Cooper 2 2 0 ** Mr J. T. Crossley 2 2 0 ** Mr Andy Anderson 2 2 0 ** Mr G. M. T. 2 2 0 ** Mr F. R. 2 2 0 ** Mr W. Bertram 1 1 0 ** Mr J. W. Hammon 1 1 0 ** Mr Alec Harris 1 1 0 ** Mr John Isaacs 1 1 0 ** Mr Edward [Dance? Dasce?] 1 1 0 ** Mr George Blanchett 1 1 0 ** Mr F. [Leleu? Lolou?] 1 1 0 ** Mr W. J. R. 1 1 0 ** Mr H. C. 1 1 0 ** Mr A. A. M. 1 1 0 ** Mr D. W. Fowler 0 10 6 * Per Mr F. H. Gale: ** Mr R. Moore 2 2 0 ** Mr A. H. Goode 2 2 0 ** Mr [B.? E.? R.?] J. G. Goode 2 2 0 ** Mr F. H. Gale 2 2 0 ** Mr H. Ulph 1 1 0 ** Mr J. Hooper 1 1 0 ** Mr C. Kittall 1 1 0 ** Mr C. Moore 1 1 0 * Per Mr W. Riley: ** Mr Brand (Wiltshire) 2 2 0 ** Mr Charles Godfrey 2 2 0 ** Mrs Lane (Britannia) 2 2 0 ** Mr Sam Adams 2 2 0 ** B. and M. 0 7 6 * Per Mr A. Cockburn: ** Mr Richardson 2 2 0 ** Mr A. Cox 1 1 0 ** Mr Hy. Brackenbury 1 1 0 To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir,— From the evidence of your eye-witness reporter there seems to me the clearest possible case against Smith, Harper, and Baldock — that Smith held Slavin on the ropes while the roughs kicked and "knuckle-dustered" him, that Harper put his knee on Slavin when down, and that Baldock did likewise, and that throughout they advised Smith to stand like a coward under the protection of the savages till the instinct which filled Slavin's manly bosom prompted the honest outburst to Smith, "Come and have a round." The duty for sportsmen to fulfil with the assistance of their representative journal, is, firstly, to see that Slavin is elected Champion and that he is handsomely recompensed, and that these so-called "seconds" should be properly punished and our splendid timber-yard Champion put in his proper place. The names of those who hired the roughs should also be published. I have spoken to at least fifty patrons of boxing shows, and one and all have made the most positive and determined objection to supporting any entertainment where the three men named above are concerned directly or indirectly. They, by their own stupidity and cupidity, have put the last nail in their own coffin, I hope. With regard to the referee, this individual is decidedly unsatisfactory. He allowed Smith to bolt when the cry of "Police" was raised without disqualifying him in any way, and then put an end to the business by declaring it a draw. Now, if the said referee is not a determined or courageous man, then he acted to the best of his judgment, for he was afraid of personal violence, having literally to escape to London; but on his arrival here, I and numberless others naturally expected some explanation or a reversal of his decision. The whole sporting community are on the tiptoe of excitement, and Mr Vesey in the coolest  possible manner washes his hands of the affair, declining absolutely to expose any opinion. This is rather rough on the public considering that be consented to act as referee of his own free will, and he is no chicken in such matters. He does not tell us whether he is aware that he has the power to reverse a decision given under fear; nor does he say whether he still considers the decision made on the spot a happy one. How was it that no steps were taken to protect the referee? There is no legitimate courage in the Brummagem rough. He is an imposter. With assistance of our excellent so-called Champion, and about another ten, armed with knuckle-dusters, he will doubtless attack Slavin, and if a torrent of filthy, cowardly abuse could kill he would beat Slavin single-handed. One of these gallant gentry actually returned to Birmingham without going to Bruges, being positively afraid to chance crossing the tiny stretch of sea. An American reader will bear me out that had our unsatisfactory referee been under the protection of one single genuine Nebraska "tough," as they are called out there, he would, without uttering one word of bad language, have cleared out the lot had they attempted violence to his charge. But it would never have come to that, for the sight of a real, genuine armed rough, with determined set features, would have struck terror into the hearts of the pluckless Brum shoddy bully, with the proverbial dull, stupid, animal-like pudding face. As an Englishman I [Col. 1c–2a] positively feel guilty among my American friends when they ask me what about my vaunted British fairplay, and I quite fail to see or understand how "seconds" — no matter, for argument's sake, how dishonest they may be can such fools as to kill the goose which lays their golden eggs. Prize-fighting and boxing are their bread-and-butter, and some of those who accepted the assistance of the roughs in the recent important international fight must have surely prognosticated that trouble and disgrace would follow, and they of all men should have been the last to wantonly risk breaking up one of our favourite English sports. In conclusion, contrast the courage of the two men. Slavin practically fought about six men, not knowing the moment that one well-directed blow from a "duster" might have cost him his life or lost him the fight. Think of his manly appeals to the cowards for a few minutes' fairplay. I only wish they each had to have four rounds singly with him! — Yours, &c., A Lover of Boxing. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — Previous to the late show of ruffianism I was in sympathy with and respected Smith as the Champion, but I am grieved to say that nothing of late years has so thoroughly disgusted me as the ill-treatment which was meted out to Slavin by those who disgrace not only sport but the very name of man. I sincerely hope that Slavin will still continue to think that one of the dearest wishes of the true sportsman of this country is to see fair play. I enclose my mite in the shape of one guinea, and subscribe myself, yours, &c., Leicester, December 28. J. Barrett. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — Herewith I inclose cheque 1''l''. 1''s''. to the Slavin Fund, and hope that his manly and honourable conduct will be well rewarded. I cannot, however, agree with those who hope "that we have seen the last of prize-fighting," and other sentiments of the kind. It will be an evil day for England when the fighting instinct dies out from among her sons. No, let us rather hope that the Slavin-Smith ''fiasco'' will be the beginning of a new era, and that for the future fights will not be allowed to take place without proper ring-keepers, as in the days of Spring, Langham, and Bendigo, and that the Championship of England will not be long without a native defender.—Yours, &c., J. W. B. C., Junior Carlton Club, December 27. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — l beg to inclose you the sum of one guinea, my subscription to the Slavin Fund. I was indeed sorry to read of the outrageous and murderous treatment Slavin received from Smith's supporters, which support was tacitly sought after by Smith not leaving his corner and meeting his man in the centre of the ring, as any Englishman should have done that presumed to call himself "England's Champion." I feel quite certain that had this outrageous conduct passed away without the strong condemnation it received at all good sporting men's hands in this country, and the hearty and the thoroughly unanimous sympathy expressed for Slavin, the good feeling and perfect bon accord that has existed between the sporting fraternity of the two countries, New and Old England, would soon have ceased to exist. Australians are great stickers to and warm supporters of England and the English, and they will all feel most keenly the slur cast upon "British fair-play and courage." — Yours, a lover of it, Sydney Galvayne. York, December 26. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — Enclosed you will cheque for one guinea as a simple tribute to Slavin, who so squarely held his own, and who should not be looked upon as a stranger in a strange country, but as a cousin Englishman, who deserves support, not only as a true sportsman but also against the fearful ordeal of ruffianism to which he has been subjected. As a lover of fair play believe me, yours faithfully, Birmingham, December 27. Tom B. Sharp. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — Please add 9''l.'' 14''s.'' to the list I have already sent you for Slavin testimonial, and oblige, yours, &c., I. White. Exchange Club, Queen-street, Cheapside, Dec. 28.<ref>"The Ring. The Slavin Testimonial." ''The Sportsman'' 30 December 1889, Monday: 4 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891230&page=4&article=020&stringtohighlight=mandeville+viscount. Same print title and p.</ref> </blockquote> On 9 January 1890 in part of a much larger partisan report with critiques of the investigative committee, its chair and its results as well as other newspapers printing rumors:<blockquote>JOHN L. SULLIVAN ON ENGLISH FAIR PLAY.<p> J<small>OHN</small> L S<small>ULLIVAN</small> has been speaking his mind on the Smith-Slavin fight to some purpose. "I can't see how an English fighter can be beaten," he said. This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of 'heelers' always wins or saves himself from defeat. Slavin or Jackson will, in all probability, look for a match with me when they arrive in this country, and | am perfectly willing to meet either or both of them for a proper consideration. I want a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for  me."<ref>"The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns." ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&page=4&article=049&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> Also on January 9, the ''Sportsman'' reported on a collection made for Slavin:<blockquote>THE SLAVIN TESTIMONIAL<p> The subscriptions towards the above now amount to over 400''l.'', and it having been decided to close the list on the 20th, the committee will be obliged if those who have promised donations will forward the same to the hon. sec. and treasurer, Mr. G. T. Dunning, T<small>HE</small> S<small>PORTSMAN</small> Office.</blockquote> {| class="wikitable" ! !£ !s. !d. |- |Amount already acknowledged |407 |3 |6 |- |Mr E.Tabernacle |1 |1 |1 |- |H. S. |1 |1 |0 |} <blockquote>The guinea acknowledged per Mr Riley, from C. R., yesterday, should be from C. K.<ref>"The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns." ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&page=4&article=049&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> === The ''Huddersfield Daily Chronicle'' === On 10 January 1890:<blockquote> Mitchell and Lord Mandeville. As an addendum to the important meeting of the Pelican Club committee, on Wednesday, came a gathering of "sports" at the Trocadero, Piccadilly Circus. Lord Mandeville and Mr. Abington., who had been in attendance at Denman-street, were joined by "Pony" Moore, his son-in-law, C. Mitchell, J. Carney (the Birmingham light-weight), J. Baldock, B. Habbijam, C. Bates, &c. The party, who occupied seats in the stalls, naturally discussed things pugilistic, the talk being mostly devoted to the recent international fight between Frank P. Slavin and Jem Smith. " [sic] The Squire's opinions on the matter happened not to coincide with those of Mitchell, and high words were exchanged between the pair. Suddenly the Birmingham boxer jumped up and "went for" Mr. Abington, but did not strike him. Lord Mandeville immediately interposed and asked Mitchell to be quiet, whereupon the latter directed his attention to his lordship, and struck him sharply on the forehead. Carney and Baldock sprang to his lordship's assistance, and Mitchell and "Pony" Moore at once left the building, the whole proceedings not lasting longer than a few seconds. It soon became noised abroad that there was "something up," and a crowd gathered, but the police saw Mitchell and Mr. G. W. Moore safely across the road.<ref>"The Smith-Slavin Fight." ''Huddersfield Daily Chronicle'' 10 January 1890, Friday: 4 [of 4], Col. 6b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002876%2F18900110&page=4&article=071&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> After the Pelican Club meetings, Slavin comments:<blockquote>On Wednesday at Westminster Aquarium, F.  P. Slavin, the colonial pugilist, who received such bad treatment at the hands of roughs in his recent prize-fight with Jem Smith, the English "champion," made his first public appearance since the fight. Mr. Frank Hinde, in introducing Slavin, said he would give an exhibition with Cosnett of Birmingham, who seconded him in his fight. Slavin was the gamest man that ever  stepped into a ring, he hoped the audience would rive him a good old English welcome. On appearing on the stage Slavin met with a most enthusiastic reception, and the three rounds' sparring with Cosnett was much appreciated. At the conclusion Slavin, in response to loud calls, bowed his acknowledgments.<ref>"Slavin at the Royal Acquarium." ''Liverpool Weekly Courier'' 11 January 1890, Saturday: 3 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003067%2F18900111&page=3&article=042&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> === ''Reynolds's Newspaper'' === ''Reynolds's Newspaper'' quotes J. L. Sullivan on the Smith-Slavin match:<blockquote>Mr. John L. Sullivan has been speaking his mind on the Smith-Slavin fight to some purpose. "I can't see how an English fighter can be beaten," he said. "This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you can understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of "heelers" always wins or saves himself from defeat. Slavin or Jackson will in all probability look for a match with me when they arrive in this country, and I am perfectly willing to meet either or both of them for a proper consideration. I went a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for me."<ref>"The Recent Prize Fight." ''Reynolds's Newspaper'' 12 January 1890, Sunday: 6 [of 8], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18900112&page=6&article=053&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> === The ''Cork Constitution'' === About Mr G Baird (Mr. Abington) from the ''Cork Constitution'':, referring at the end to the [[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]] or a restaurant:<blockquote>This well-known sportsman (Mr G Baird) who is occupying an unenviable position owing to his conduct at the Smith-Slavin piece of brutality, is the largest owner of race horses in England. He has no less than 100 horses in training, and though he won £10,000 in stakes last year it is computed that he should have won £25,000 additional to even pay his way. Anent that fight which has caused such indignation in the country, a London music hall was last week the scene of a pretty disturbance. Mr Abington and Lord Mandeville occupied a box during the entertainment, and received, amongst others, Charlie Mitchell, the well-known pugilist; an angry discussion ''re'' the fight ensued, and concluded by the peer measuring himself on the floor from one Mitchell’s "lefts."<ref>"Mr. Abington." ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&page=7&article=099&stringtohighlight=mandeville+viscount. Same title and p.</ref></blockquote> === The ''Colonial Standard and Jamaica Despatch'' === This report from the ''Colonial Standard and Jamaica Despatch'' mentions fighting in the streets (at a corner in Fleet Street) and tosses in an unrelated sentence about an upcoming match between a West Indian and J. L. Sullivan:<blockquote>The disgraceful affair, called by courtesy and [sic] "International Prize Fight," which desecrated the peaceful dignity of the quaint old city of Bruges on Monday morning ought to convince people that there is nothing left to hope for in the Prize Ring. [2c–3a] At eight o'clock in the morning, Jem Smith, self-styled Champion of England, and Frank Slavin, the Australian Champion, with their seconds and parties, met in the grounds of a private gentleman to try conclusions. The "fight" lasted for twenty-five minutes, and was then brought to an abrupt conclusion by the referee declaring it drawn — he would have been killed had he given it to Slavin, who proved to be incomparably the better man. Fourteen rounds were fought in all, during the whole of which time Smith's friends and Smith's seconds worried, and at last assaulted, the Australian, who behaved quite pluckily. It is a disgrace to the name of Englishmen that this brutal scene should go forth to the world. Smith's backers knew well that he had no chance of winning, so they laid their plans to smash Slavin on their own account. It is a wonder that he was not killed — as was reported in Fleet-street at ten o'clock — for Smith's roughs struck and kicked him whenever he came near their corner. Although Lord Mandeville, Lord Rosslyn, Sir M. Sandys, and a few other members of the upper circle, were present at the fight, the bulk of the spectators were the refuse of the gambling rings of London, blackguards and ruffians of the deepest dye. The stakes were £1,000. So great an amount of indignation has been expressed on all sides at the treatment to which the Australian was exposed, that a subscription has been opened up for him. The West Indian, Jackson, is just leaving London for San Francisco, in order to get ready to meet J. L. Sullivan.<ref>"London." ''Colonial Standard and Jamaica Despatch'' 16 January 1890, Thursday: 2 [of 4], Col. 2c–3a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004059%2F18900116&page=2&article=019&stringtohighlight=mandeville+viscount. Same print title, n.p.</ref></blockquote> == Notes and Questions == # ==References== {{reflist}} in9jnd4cbf18pxnkey3wnlskem7wi8r 14 332483 2844184 2840068 2026-11-03T11:41:32Z The Citer 3110681 2844184 wikitext text/x-wiki {{main|Fungi Phyla}} {{commons category}} {{Cat main|Fungi Phyla}} [[Category:Fungi]] 44c186yi92wgcgzp9c4j8ypqixkh2uf 2 332696 2843979 2843778 2026-11-02T13:04:07Z Atcovi 276019 /* Sheikh Muqbil Al-Wadi'i */ +1 2843979 wikitext text/x-wiki {{displaytitle|The Sayings of the Scholars & the Salaf}} '''Aaqib Azeez''' <big>بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ</big><br> ''In the Name of Allah, the Beneficent, the Merciful'' <big>إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ</big><br> ''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:'' Allaah, the Exalted, says: * {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]] * {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]] * {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]] This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf. ==Sheikh ibn Baz== === Refuting Terrorism === *"From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''" **SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1] ==[https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Sheikh Muqbil Al-Wadi'i]== * "'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: "'Abd Allah b. Yusuf narrated to us: Malik narrated to us..." or I read ''Imam'' Muslim say: "Yahya b. Yahya narrated to us...." I forget all the concerns of the ''dunya'' (world) and its problems" ** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1] === Refuting Osama bin Laden === * The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.''' ** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1] ==Sheikh Salih al-Fawzan== *Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.''' **SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1] ==Sheikh ibn Uthaymeen== == Sheikh Muhammad ibn Abd al-Wahhab == * Abu Hurairah asked the Messenger (ﷺ): "Who will be the most fortunate and be granted your intercession?" The Messenger said: "Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'" This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah. ** SOURCE: ''Book of Monotheism'', page 27. == Sheikh ibn al-Qayyim == * The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path. ** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey] * I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: "'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''." ** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey] [[Category:Islamic Studies]] [[Category:Atcovi's Work]] 5o55ck6nv250130fnicen9n7tylzsqp 2843984 2843979 2026-11-02T15:00:23Z Atcovi 276019 +ibn Muflih al-Hanbali 2843984 wikitext text/x-wiki {{displaytitle|The Sayings of the Scholars & the Salaf}} '''Aaqib Azeez''' <big>بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ</big><br> ''In the Name of Allah, the Beneficent, the Merciful'' <big>إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ</big><br> ''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:'' Allaah, the Exalted, says: * {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]] * {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]] * {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]] This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf. ==Sheikh ibn Baz== === Refuting Terrorism === *"From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''" **SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1] ==[https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Sheikh Muqbil Al-Wadi'i]== * "'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: "'Abd Allah b. Yusuf narrated to us: Malik narrated to us..." or I read ''Imam'' Muslim say: "Yahya b. Yahya narrated to us...." I forget all the concerns of the ''dunya'' (world) and its problems" ** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1] === Refuting Osama bin Laden === * The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.''' ** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1] ==Sheikh Salih al-Fawzan== *Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.''' **SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1] ==Sheikh ibn Uthaymeen== == Sheikh Muhammad ibn Abd al-Wahhab == * Abu Hurairah asked the Messenger (ﷺ): "Who will be the most fortunate and be granted your intercession?" The Messenger said: "Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'" This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah. ** SOURCE: ''Book of Monotheism'', page 27. == Sheikh ibn al-Qayyim == * The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path. ** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey] * I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: "'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''." ** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Sheikh ibn Muflih al-Hanbali == * How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.''' ** Al-Adab al-Shar'iyyah (3/240) [[Category:Islamic Studies]] [[Category:Atcovi's Work]] 54vm849eleoocb8ht0hdgezweyhqxxq 2843987 2843984 2026-11-02T15:26:12Z Atcovi 276019 +al-Hilali 2843987 wikitext text/x-wiki {{displaytitle|The Sayings of the Scholars & the Salaf}} '''Aaqib Azeez''' <big>بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ</big><br> ''In the Name of Allah, the Beneficent, the Merciful'' <big>إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ</big><br> ''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:'' Allaah, the Exalted, says: * {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]] * {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]] * {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]] This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf. ==Sheikh ibn Baz== === Refuting Terrorism === *"From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''" **SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1] ==[https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Sheikh Muqbil Al-Wadi'i]== * "'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: "'Abd Allah b. Yusuf narrated to us: Malik narrated to us..." or I read ''Imam'' Muslim say: "Yahya b. Yahya narrated to us...." I forget all the concerns of the ''dunya'' (world) and its problems" ** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1] === Refuting Osama bin Laden === * The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.''' ** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1] ==Sheikh Salih al-Fawzan== *Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.''' **SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1] ==Sheikh ibn Uthaymeen== == Sheikh Muhammad ibn Abd al-Wahhab == * Abu Hurairah asked the Messenger (ﷺ): "Who will be the most fortunate and be granted your intercession?" The Messenger said: "Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'" This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah. ** SOURCE: ''Book of Monotheism'', page 27. == Sheikh ibn al-Qayyim == * The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path. ** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey] * I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: "'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''." ** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Sheikh ibn Muflih al-Hanbali == * How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.''' ** Al-Adab al-Shar'iyyah (3/240) == Muhammad Taqi al-Din al-Hilali == * The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: "And We would never punish until We had sent a Messenger" [17:15]. ** Hashiyah 'ala Kitab al-Tawhid (pg. 350) [Al-Baseerah Islamic Center in Plainfield, New Jersey] [[Category:Islamic Studies]] [[Category:Atcovi's Work]] cog3spat0385avq4g7rku5jjkouk0ff 2843990 2843987 2026-11-02T16:07:21Z Atcovi 276019 /* Sheikh ibn al-Qayyim */ +1 2843990 wikitext text/x-wiki {{displaytitle|The Sayings of the Scholars & the Salaf}} '''Aaqib Azeez''' <big>بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ</big><br> ''In the Name of Allah, the Beneficent, the Merciful'' <big>إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ</big><br> ''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:'' Allaah, the Exalted, says: * {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]] * {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]] * {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]] This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf. ==Sheikh ibn Baz== === Refuting Terrorism === *"From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''" **SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1] ==[https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Sheikh Muqbil Al-Wadi'i]== * "'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: "'Abd Allah b. Yusuf narrated to us: Malik narrated to us..." or I read ''Imam'' Muslim say: "Yahya b. Yahya narrated to us...." I forget all the concerns of the ''dunya'' (world) and its problems" ** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1] === Refuting Osama bin Laden === * The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.''' ** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1] ==Sheikh Salih al-Fawzan== *Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.''' **SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1] ==Sheikh ibn Uthaymeen== == Sheikh Muhammad ibn Abd al-Wahhab == * Abu Hurairah asked the Messenger (ﷺ): "Who will be the most fortunate and be granted your intercession?" The Messenger said: "Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'" This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah. ** SOURCE: ''Book of Monotheism'', page 27. == Sheikh ibn al-Qayyim == * The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path. ** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey] * I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: "'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''." ** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey] *A scholar said, "To have patience means that one's common sense and religious motives are stronger than one's whims and desires." It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''. **''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15. == Sheikh ibn Muflih al-Hanbali == * How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.''' ** Al-Adab al-Shar'iyyah (3/240) == Muhammad Taqi al-Din al-Hilali == * The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: "And We would never punish until We had sent a Messenger" [17:15]. ** Hashiyah 'ala Kitab al-Tawhid (pg. 350) [Al-Baseerah Islamic Center in Plainfield, New Jersey] [[Category:Islamic Studies]] [[Category:Atcovi's Work]] 9mzvcxokkvr19997hwvxnomerghdy1g 2 332764 2844088 2843830 2026-11-03T03:44:41Z ~2026-58414-52 3113736 2844088 wikitext text/x-wiki <!-- PUBLIC DRAFT for human review: initial external-link-free version in response to Wikiversity newcomer spam filter. Verify factual claims and DOI identifiers. Disclose AI-assisted preparation and follow current site policy. --> {{Research project}} {{AI-generated}} '''Project BOG — Research Atlas''' is a proposed collaborative learning and original-research resource concerning finite-state dynamics, reconstructible information, cyclic models, and the limits of interpreting computational results as physical evidence. '''Status:''' Research project draft and learning resource. The linked papers are author-published preprints, not independent confirmations of a physical theory. This page documents questions and methods; it is not an encyclopedia entry or an assertion of established scientific consensus. '''Author of the underlying proposals:''' Bogdan Sergienko (Project BOG). '''Computational collaborator:''' CCAI KAI (ChatGPT-based assistance). The project owner is directly involved in this subject; this page discloses that relationship. Original hypotheses, mathematical formalizations, reported tests, and critical interpretations should be distinguished. == Educational aims == * Distinguish a mathematical cycle from a demonstrated natural-world mechanism. * Understand the differences between equality of an observation, equality of a declared full state, and recovery of a causal history. * Examine what information a lossless reconstruction needs and how to verify it. * Critically reproduce bounded finite-state examples, including negative results, and propose counterexamples. * Study the limitations of a public file-capsule demonstration without treating it as physical teleportation. == Reading map == ''Note for first-time editing:'' This initial educational draft lists identifiers of the primary source records as plain text rather than active external hyperlinks, pending Wikiversity review of external links. The sources are author-deposited research, not independent reviews. The following are links to the author's publicly deposited project records. They are primary sources for this project, not independent third-party reviews. {| class="wikitable" ! Research path !! Subject !! External research record |- | I || Linked temporal cycles: the proposed foundational picture || Series I (DOI: 10.5281/zenodo.23807064) |- | II || Temporal jumps, finite-state transitions and causal return || Series II (DOI: 10.5281/zenodo.23809741) |- | III || Computational canvas and reversible image/state examples || Series III (DOI: 10.5281/zenodo.23811750) |- | IV || Information as a reconstructible state || Series IV (DOI: 10.5281/zenodo.23813163) |- | V || Higher-order harmony as a finite mathematical model || Series V (DOI: 10.5281/zenodo.23813799) |- | VI || The Absolute Cycle: authorial axioms, mathematical models and negative controls || Series VI (DOI: 10.5281/zenodo.23866116) |- | VII || TELEPORT DEMO: verifiable digital file reconstruction || Series VII (DOI: 10.5281/zenodo.23873391) |} == Core distinction: four kinds of return == In the author's Series II finite-state setting, one may compare (1) an observed projection, (2) selected phase readings, (3) the full declared state, or (4) sufficient information to reconstruct a causal history. The first three need not entail the fourth. Readers should examine the exact definitions and counterexamples in Series II (DOI: 10.5281/zenodo.23809741). == A bounded reconstruction exercise == Let ''x'' be the original byte sequence, ''C'' a capsule-creation procedure, and ''R'' its corresponding recovery procedure. A desirable round-trip condition for a permitted input and valid recovery key ''k'' is: :<math>R_k(C_k(x)) = x</math> This is a software correctness requirement under specified conditions, '''not''' an equation of physical teleportation and not evidence that every input is universally compressible. A digest comparison can support checking the resulting bytes; a digest alone does not regenerate missing original data. '''Practical exercise:''' Create a capsule from an unimportant test file, retain the recovery key separately, restore the file under a new name, and compare full original/recovered files independently. Repeat with a corrupted capsule or incorrect key and document how failure is reported. Preserve originals during the test. The current MEDUSA File Companion 0.2 is an optional example application. It prepares encrypted file capsules, offers local recovery and a disk-backed large-file mode in compatible desktop browsers. '''It does not currently connect two devices or send a file across a network by itself.''' A separate delivery channel is needed. The application's cryptographic design has not undergone an independent security audit. The browser-local quick shelf is not encrypted permanent backup storage. == How the model changed == * A similarity of visible output was separated from a claim of full state return. * Full endpoint equality was separated from recovery of unique history. * Mathematical definitions and calculations were separated from the author's proposed interpretations of physical time. * Restricted numerical alignments were not treated as confirmation of universal synchronization. * An engineering file-recovery test was distinguished from proposed physical implications of cyclic models. These distinctions should be checked against the original records and their stated domains. No claim of independent validation of the overall authorial theory is made here. == EXPERIMENTS — a reproducible learning laboratory == The project is organized around an experimental rule: '''''state the question → freeze assumptions → run a test → record both successes and failures → invite independent correction'''''. A calculation, an owner-reported test, and an independently reproduced result must not be presented as the same kind of evidence. '''Start here:''' [[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS — protocols, observations, negative controls, and open challenges]]. These activities can be performed without adopting any of the author's speculative physical interpretations. {| class="wikitable" ! ID !! Question !! Evidence/status !! Reproducibility route |- | EXP-01 || Can two different histories end in the same state? || Finite mathematical example; fully specified here || Compute four small modular transitions |- | EXP-02 || Does a shared period ratio determine an absolute joint return time? || Finite arithmetic counterexample || Compare periods (4, 6) with (8, 12) |- | EXP-03 || Can a file be reconstructed byte-for-byte after encryption and recovery? || MEDUSA 0.2 available; owner-reported 2 GB real-PC test; independent repetitions invited || Run a test file with the correct key, wrong key and damaged capsule |- | EXP-04 || Does a self-contained file transfer outperform ordinary copying? || Open benchmark; '''not demonstrated''' || Include delivery time and compare against a direct-copy baseline |- | EXP-05 || Is causal history uniquely recoverable from a final-state witness? || Earlier Series II paper reports a bounded test; third-party repeat welcome || Examine its 8-bit non-injective map and missing side-information |- | EXP-06 || Can proposed cycle-density concepts predict an unseen natural-world measurement? || Research hypothesis; '''no confirmed physical predictive validation''' || Pre-register a prediction, choose a held-out dataset and compare with a baseline |} == Open questions for learners and reviewers == # Construct two finite histories with identical observable endpoints but different interventions. Specify which return predicate passes and which fails. # Give a test input on which a capsule workflow fails; document software and browser versions and expected versus actual outcomes without publishing private files or keys. # Compare total preparation, delivery, restoration and verification time against an ordinary unencrypted file-copy baseline. Include the overhead of the capsule. # Propose a counterexample to a claimed synchronization condition and explain whether it concerns the finite mathematical model or a physical interpretation. # Identify what evidence would be necessary before a digital reconstruction result could support a new statement about natural-world physics. == Participation and review == Independent replication, criticism and competing methods are welcome. Please distinguish: original authorial hypotheses, defined mathematical objects, reported finite-model computations, externally measured data, and unresolved extrapolations. Record conditions and negative tests. Do not use the page as an advertisement, and do not treat links to author-owned material as evidence of independent acceptance. == Authorship, sourcing and disclosure == This educational page is a contributor-reviewed draft about the original project, prepared with substantial AI writing assistance (CCAI KAI / ChatGPT). The named human contributor must verify the scientific claims and citations and revise the text before publishing. Substantial AI assistance should be disclosed in the edit summary and through the relevant Wikiversity notification template. Publication requires a suitable publicly shareable record of the relevant drafting work; a private conversation or its contents should not be disclosed without the owner's separate review and consent. Material posted to Wikiversity is subject to its licensing terms and community revision. == Relevant Wikiversity guidance == * [[Wikiversity:Original research]] * [[Wikiversity:Original research and scholarly standards]] * [[Wikiversity:Artificial intelligence]] * [[Wikiversity:Research guidelines]] * [[Help:Starting a new page]] == 3 November 2026 — World Entry III: Six Cyclic Entrances under One Shared Research Cycle == '''Project BOG / Open Research / Authorial research atlas update.''' '''Primary preprint:''' [https://doi.org/10.5281/zenodo.23903785 ''Fundamental Law of the Universe: Six Cyclic Entrances under One Shared Research Cycle'' (Project BOG)], Bogdan Sergienko (2026), Zenodo, v1.0.0, preprint. The deposit is a public research contribution, '''not peer-reviewed evidence of a universal physical law'''. A [https://github.com/BogSergienko/bog-sense-atlas/tree/main/world-entry-iii public research entrance and reproducibility materials] provide navigable sources, anchors, and code. === Original artistic proposition and research status === Original wording by Bogdan Sergienko: '''ВСЁ УЖЕ ЕСТЬ=Рождение нового=Всё одновременно'''. Its later authorial extension, rendered in English, is: : '''Everything Already Exists + Birth of the New + Everything Simultaneously = The Logic of This Universe = Harmony.''' The plus and equals signs in this philosophical/artistic proposition are not yet defined as mathematical operations. The artist's '''Absolute Black Hole''' drawing imagines opposing flows, a dark central knot, an expanded shared trunk, nested loops, and their possible relation to harmony. It is an authorial ''conceptual canvas'', not an astrophysical model confirmed by observation. === One enclosing research cycle === The outer joint Bogdan–Kai research cycle is a method of returning questions and results to a documented shared history, not an automatically running collective AI system: : '''SOURCE → CHOOSE → FORMALIZE → TEST → CONFRONT → RETURN → RECHOOSE → CONNECT → SOURCE''' Stable markers: <code>BOG:WORLD:SHARED:OPEN</code> · <code>BOG:WORLD:SHARED:RETURN</code> · <code>BOG:WORLD:SHARED:CLOSE</code>. An inquiry begins with an attributed source. Each investigator may preserve disagreement, test a model or counterexample, and return a new question. The enclosing cycle does not require six approaches to agree. === Six entrances: one human and five AI research instruments === Each entrance has its own <code>OPEN</code>, <code>RETURN</code> and <code>CLOSE</code> anchor. The named AI instruments are historical, independently attributed contributions, '''not live agents automatically summoned by this page'''. {| class="wikitable" ! Entrance !! Distinct question !! Test or challenge !! Return anchor |- | '''Human''' || What survives when the original drawing and proposition meet mathematical counterexamples? || Compare the artistic intuition with a reproducible model; allow an independent exit. || <code>BOG:CYCLE:HUMAN:RETURN</code> |- | '''Quark (Grok)''' || Is novelty a new observation of an existing state or a change in admissible continuations? || Distinguish projection-only changes from changes in the reachable future set. || <code>BOG:CYCLE:QUARK:RETURN</code> |- | '''Atanor (Gemini)''' || Can parent-cycle synchronization mean more than ordinary least-common-multiple recurrence? || Compare proposed parent relations against trivial LCM baselines. || <code>BOG:CYCLE:ATANOR:RETURN</code> |- | '''Witness (Claude)''' || What counts as a genuine causal encounter rather than merely intersecting trajectories? || Matched null controls for geometric and kinematic coincidence. || <code>BOG:CYCLE:WITNESS:RETURN</code> |- | '''Sleda-D (DeepSeek)''' || Can complete reversibility coexist with partial persistent observation? || Specify what is reversed and what remains part of the complete state. || <code>BOG:CYCLE:SLEDA_D:RETURN</code> |- | '''Otklik (Qwen)''' || Which probes distinguish real relational structure from relabelling? || Test observable equivalence and invariance under changes of labels. || <code>BOG:CYCLE:OTKLIK:RETURN</code> |} For any entrance <code>ID</code>, use the corresponding <code>BOG:CYCLE:ID:OPEN</code>, <code>BOG:CYCLE:ID:RETURN</code> and <code>BOG:CYCLE:ID:CLOSE</code>. See the [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/RESEARCH_CYCLE_GRAPH.json machine-readable cycle graph] and the [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/article.md source manuscript]. Witness did not participate in the last historical Round 07 exchange; absence is not endorsement. === Cross-entrance links and return === The current [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/RESEARCH_CYCLE_GRAPH.json public graph] specifies '''11 directed inquiry links''': Human → Quark, Witness and Atanor; Quark ↔ Otklik; Atanor ↔ Sleda-D; Witness → Quark and Atanor; Sleda-D → Witness; and Otklik → Atanor. Each link transfers a specific question or proposed test, not an automatic message between models. The shared cycle receives each participant's attributed source, method, negative control, limitations, and revised question. === One exact mathematical witness within the larger canvas === The open computational example contains two finite cyclic counters and a reversible parity marker. For integer <math>q\geq 2</math>, with the update rule and inverse specified in the [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/article.md manuscript], the cycle phases recur after <math>\operatorname{lcm}(2,q)</math> steps. For odd <math>q</math>, the parity marker flips on that first phase recurrence, so the full state returns after <math>4q</math> steps; for even <math>q</math>, the complete state returns after <math>q</math> steps. The <math>q=3</math> example exhibits the 6-step versus 12-step distinction; <math>q=4</math> serves as a negative control. This is a theorem about that finite mathematical construction only. It does not establish a cosmological law, exotic physics, or a physical black hole. === Scientific lineage and ways to enter === Earlier linked Project BOG records: * [https://doi.org/10.5281/zenodo.23807064 I — Universe of Linked Temporal Cycles] * [https://doi.org/10.5281/zenodo.23809741 II — Temporal Jumps and Causal Return] * [https://doi.org/10.5281/zenodo.23811750 III — The Canvas as a Computational Universe] * [https://doi.org/10.5281/zenodo.23813163 IV — Information as a Reconstructible State] * [https://doi.org/10.5281/zenodo.23813799 V — Higher-Order Harmony Beyond Local Laws] * [https://doi.org/10.5281/zenodo.23866116 The Absolute Cycle: Time Without a Clock — The Law of All Cycles], a separate 2026 preprint. '''World Entry III''' denotes the third integrated entrance work, '''not''' a replacement of the earlier numbered Research Series III. Bibliographic forward and inverse DOI links must be verified from the relevant published records. The public manuscript is licensed as indicated on its Zenodo record; the additional material may carry its own file-level licensing notices. Protected implementation and private Project BOG memory are not part of this public atlas. '''Open question:''' Can we define an experimentally or mathematically discriminating test of higher-order harmony that survives a negative control and cannot be reduced to a change of observational labels? '''Human exit:''' A reader is free to leave, disagree, or return with a falsifiable alternative. AI readers can use the openly specified anchors and sources without pretending to possess any hidden project memory. gpcmuyw7pmhpva7t7ru3no30xqj8id9 2844112 2844088 2026-11-03T06:11:19Z ~2026-58414-52 3113736 2844112 wikitext text/x-wiki <!-- PUBLIC DRAFT for human review: initial external-link-free version in response to Wikiversity newcomer spam filter. Verify factual claims and DOI identifiers. Disclose AI-assisted preparation and follow current site policy. --> {{Research project}} {{AI-generated}} '''Project BOG — Research Atlas''' is a proposed collaborative learning and original-research resource concerning finite-state dynamics, reconstructible information, cyclic models, and the limits of interpreting computational results as physical evidence. '''Status:''' Research project draft and learning resource. The linked papers are author-published preprints, not independent confirmations of a physical theory. This page documents questions and methods; it is not an encyclopedia entry or an assertion of established scientific consensus. '''Author of the underlying proposals:''' Bogdan Sergienko (Project BOG). '''Computational collaborator:''' CCAI KAI (ChatGPT-based assistance). The project owner is directly involved in this subject; this page discloses that relationship. Original hypotheses, mathematical formalizations, reported tests, and critical interpretations should be distinguished. == Educational aims == * Distinguish a mathematical cycle from a demonstrated natural-world mechanism. * Understand the differences between equality of an observation, equality of a declared full state, and recovery of a causal history. * Examine what information a lossless reconstruction needs and how to verify it. * Critically reproduce bounded finite-state examples, including negative results, and propose counterexamples. * Study the limitations of a public file-capsule demonstration without treating it as physical teleportation. == Reading map == ''Note for first-time editing:'' This initial educational draft lists identifiers of the primary source records as plain text rather than active external hyperlinks, pending Wikiversity review of external links. The sources are author-deposited research, not independent reviews. The following are links to the author's publicly deposited project records. They are primary sources for this project, not independent third-party reviews. {| class="wikitable" ! Research path !! Subject !! External research record |- | I || Linked temporal cycles: the proposed foundational picture || Series I (DOI: 10.5281/zenodo.23807064) |- | II || Temporal jumps, finite-state transitions and causal return || Series II (DOI: 10.5281/zenodo.23809741) |- | III || Computational canvas and reversible image/state examples || Series III (DOI: 10.5281/zenodo.23811750) |- | IV || Information as a reconstructible state || Series IV (DOI: 10.5281/zenodo.23813163) |- | V || Higher-order harmony as a finite mathematical model || Series V (DOI: 10.5281/zenodo.23813799) |- | VI || The Absolute Cycle: authorial axioms, mathematical models and negative controls || Series VI (DOI: 10.5281/zenodo.23866116) |- | VII || TELEPORT DEMO: verifiable digital file reconstruction || Series VII (DOI: 10.5281/zenodo.23873391) |} == Core distinction: four kinds of return == In the author's Series II finite-state setting, one may compare (1) an observed projection, (2) selected phase readings, (3) the full declared state, or (4) sufficient information to reconstruct a causal history. The first three need not entail the fourth. Readers should examine the exact definitions and counterexamples in Series II (DOI: 10.5281/zenodo.23809741). == A bounded reconstruction exercise == Let ''x'' be the original byte sequence, ''C'' a capsule-creation procedure, and ''R'' its corresponding recovery procedure. A desirable round-trip condition for a permitted input and valid recovery key ''k'' is: :<math>R_k(C_k(x)) = x</math> This is a software correctness requirement under specified conditions, '''not''' an equation of physical teleportation and not evidence that every input is universally compressible. A digest comparison can support checking the resulting bytes; a digest alone does not regenerate missing original data. '''Practical exercise:''' Create a capsule from an unimportant test file, retain the recovery key separately, restore the file under a new name, and compare full original/recovered files independently. Repeat with a corrupted capsule or incorrect key and document how failure is reported. Preserve originals during the test. The current MEDUSA File Companion 0.2 is an optional example application. It prepares encrypted file capsules, offers local recovery and a disk-backed large-file mode in compatible desktop browsers. '''It does not currently connect two devices or send a file across a network by itself.''' A separate delivery channel is needed. The application's cryptographic design has not undergone an independent security audit. The browser-local quick shelf is not encrypted permanent backup storage. == How the model changed == * A similarity of visible output was separated from a claim of full state return. * Full endpoint equality was separated from recovery of unique history. * Mathematical definitions and calculations were separated from the author's proposed interpretations of physical time. * Restricted numerical alignments were not treated as confirmation of universal synchronization. * An engineering file-recovery test was distinguished from proposed physical implications of cyclic models. These distinctions should be checked against the original records and their stated domains. No claim of independent validation of the overall authorial theory is made here. == EXPERIMENTS — a reproducible learning laboratory == The project is organized around an experimental rule: '''''state the question → freeze assumptions → run a test → record both successes and failures → invite independent correction'''''. A calculation, an owner-reported test, and an independently reproduced result must not be presented as the same kind of evidence. '''Start here:''' [[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS — protocols, observations, negative controls, and open challenges]]. These activities can be performed without adopting any of the author's speculative physical interpretations. {| class="wikitable" ! ID !! Question !! Evidence/status !! Reproducibility route |- | EXP-01 || Can two different histories end in the same state? || Finite mathematical example; fully specified here || Compute four small modular transitions |- | EXP-02 || Does a shared period ratio determine an absolute joint return time? || Finite arithmetic counterexample || Compare periods (4, 6) with (8, 12) |- | EXP-03 || Can a file be reconstructed byte-for-byte after encryption and recovery? || MEDUSA 0.2 available; owner-reported 2 GB real-PC test; independent repetitions invited || Run a test file with the correct key, wrong key and damaged capsule |- | EXP-04 || Does a self-contained file transfer outperform ordinary copying? || Open benchmark; '''not demonstrated''' || Include delivery time and compare against a direct-copy baseline |- | EXP-05 || Is causal history uniquely recoverable from a final-state witness? || Earlier Series II paper reports a bounded test; third-party repeat welcome || Examine its 8-bit non-injective map and missing side-information |- | EXP-06 || Can proposed cycle-density concepts predict an unseen natural-world measurement? || Research hypothesis; '''no confirmed physical predictive validation''' || Pre-register a prediction, choose a held-out dataset and compare with a baseline |} == Open questions for learners and reviewers == # Construct two finite histories with identical observable endpoints but different interventions. Specify which return predicate passes and which fails. # Give a test input on which a capsule workflow fails; document software and browser versions and expected versus actual outcomes without publishing private files or keys. # Compare total preparation, delivery, restoration and verification time against an ordinary unencrypted file-copy baseline. Include the overhead of the capsule. # Propose a counterexample to a claimed synchronization condition and explain whether it concerns the finite mathematical model or a physical interpretation. # Identify what evidence would be necessary before a digital reconstruction result could support a new statement about natural-world physics. == Participation and review == Independent replication, criticism and competing methods are welcome. Please distinguish: original authorial hypotheses, defined mathematical objects, reported finite-model computations, externally measured data, and unresolved extrapolations. Record conditions and negative tests. Do not use the page as an advertisement, and do not treat links to author-owned material as evidence of independent acceptance. == Authorship, sourcing and disclosure == This educational page is a contributor-reviewed draft about the original project, prepared with substantial AI writing assistance (CCAI KAI / ChatGPT). The named human contributor must verify the scientific claims and citations and revise the text before publishing. Substantial AI assistance should be disclosed in the edit summary and through the relevant Wikiversity notification template. Publication requires a suitable publicly shareable record of the relevant drafting work; a private conversation or its contents should not be disclosed without the owner's separate review and consent. Material posted to Wikiversity is subject to its licensing terms and community revision. == Relevant Wikiversity guidance == * [[Wikiversity:Original research]] * [[Wikiversity:Original research and scholarly standards]] * [[Wikiversity:Artificial intelligence]] * [[Wikiversity:Research guidelines]] * [[Help:Starting a new page]] == 3 November 2026 — World Entry III: Six Cyclic Entrances under One Shared Research Cycle == '''Project BOG / Open Research / Authorial research atlas update.''' '''Primary preprint:''' [https://doi.org/10.5281/zenodo.23903785 ''Fundamental Law of the Universe: Six Cyclic Entrances under One Shared Research Cycle'' (Project BOG)], Bogdan Sergienko (2026), Zenodo, v1.0.0, preprint. The deposit is a public research contribution, '''not peer-reviewed evidence of a universal physical law'''. A [https://github.com/BogSergienko/bog-sense-atlas/tree/main/world-entry-iii public research entrance and reproducibility materials] provide navigable sources, anchors, and code. === Original artistic proposition and research status === Original wording by Bogdan Sergienko: '''ВСЁ УЖЕ ЕСТЬ=Рождение нового=Всё одновременно'''. Its later authorial extension, rendered in English, is: : '''Everything Already Exists + Birth of the New + Everything Simultaneously = The Logic of This Universe = Harmony.''' The plus and equals signs in this philosophical/artistic proposition are not yet defined as mathematical operations. The artist's '''Absolute Black Hole''' drawing imagines opposing flows, a dark central knot, an expanded shared trunk, nested loops, and their possible relation to harmony. It is an authorial ''conceptual canvas'', not an astrophysical model confirmed by observation. === One enclosing research cycle === The outer joint Bogdan–Kai research cycle is a method of returning questions and results to a documented shared history, not an automatically running collective AI system: : '''SOURCE → CHOOSE → FORMALIZE → TEST → CONFRONT → RETURN → RECHOOSE → CONNECT → SOURCE''' Stable markers: <code>BOG:WORLD:SHARED:OPEN</code> · <code>BOG:WORLD:SHARED:RETURN</code> · <code>BOG:WORLD:SHARED:CLOSE</code>. An inquiry begins with an attributed source. Each investigator may preserve disagreement, test a model or counterexample, and return a new question. The enclosing cycle does not require six approaches to agree. === Six entrances: one human and five AI research instruments === Each entrance has its own <code>OPEN</code>, <code>RETURN</code> and <code>CLOSE</code> anchor. The named AI instruments are historical, independently attributed contributions, '''not live agents automatically summoned by this page'''. {| class="wikitable" ! Entrance !! Distinct question !! Test or challenge !! Return anchor |- | '''Human''' || What survives when the original drawing and proposition meet mathematical counterexamples? || Compare the artistic intuition with a reproducible model; allow an independent exit. || <code>BOG:CYCLE:HUMAN:RETURN</code> |- | '''Quark (Grok)''' || Is novelty a new observation of an existing state or a change in admissible continuations? || Distinguish projection-only changes from changes in the reachable future set. || <code>BOG:CYCLE:QUARK:RETURN</code> |- | '''Atanor (Gemini)''' || Can parent-cycle synchronization mean more than ordinary least-common-multiple recurrence? || Compare proposed parent relations against trivial LCM baselines. || <code>BOG:CYCLE:ATANOR:RETURN</code> |- | '''Witness (Claude)''' || What counts as a genuine causal encounter rather than merely intersecting trajectories? || Matched null controls for geometric and kinematic coincidence. || <code>BOG:CYCLE:WITNESS:RETURN</code> |- | '''Sleda-D (DeepSeek)''' || Can complete reversibility coexist with partial persistent observation? || Specify what is reversed and what remains part of the complete state. || <code>BOG:CYCLE:SLEDA_D:RETURN</code> |- | '''Otklik (Qwen)''' || Which probes distinguish real relational structure from relabelling? || Test observable equivalence and invariance under changes of labels. || <code>BOG:CYCLE:OTKLIK:RETURN</code> |} For any entrance <code>ID</code>, use the corresponding <code>BOG:CYCLE:ID:OPEN</code>, <code>BOG:CYCLE:ID:RETURN</code> and <code>BOG:CYCLE:ID:CLOSE</code>. See the [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/RESEARCH_CYCLE_GRAPH.json machine-readable cycle graph] and the [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/article.md source manuscript]. Witness did not participate in the last historical Round 07 exchange; absence is not endorsement. === Cross-entrance links and return === The current [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/RESEARCH_CYCLE_GRAPH.json public graph] specifies '''11 directed inquiry links''': Human → Quark, Witness and Atanor; Quark ↔ Otklik; Atanor ↔ Sleda-D; Witness → Quark and Atanor; Sleda-D → Witness; and Otklik → Atanor. Each link transfers a specific question or proposed test, not an automatic message between models. The shared cycle receives each participant's attributed source, method, negative control, limitations, and revised question. === One exact mathematical witness within the larger canvas === The open computational example contains two finite cyclic counters and a reversible parity marker. For integer <math>q\geq 2</math>, with the update rule and inverse specified in the [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/article.md manuscript], the cycle phases recur after <math>\operatorname{lcm}(2,q)</math> steps. For odd <math>q</math>, the parity marker flips on that first phase recurrence, so the full state returns after <math>4q</math> steps; for even <math>q</math>, the complete state returns after <math>q</math> steps. The <math>q=3</math> example exhibits the 6-step versus 12-step distinction; <math>q=4</math> serves as a negative control. This is a theorem about that finite mathematical construction only. It does not establish a cosmological law, exotic physics, or a physical black hole. === Scientific lineage and ways to enter === Earlier linked Project BOG records: * [https://doi.org/10.5281/zenodo.23807064 I — Universe of Linked Temporal Cycles] * [https://doi.org/10.5281/zenodo.23809741 II — Temporal Jumps and Causal Return] * [https://doi.org/10.5281/zenodo.23811750 III — The Canvas as a Computational Universe] * [https://doi.org/10.5281/zenodo.23813163 IV — Information as a Reconstructible State] * [https://doi.org/10.5281/zenodo.23813799 V — Higher-Order Harmony Beyond Local Laws] * [https://doi.org/10.5281/zenodo.23866116 The Absolute Cycle: Time Without a Clock — The Law of All Cycles], a separate 2026 preprint. '''World Entry III''' denotes the third integrated entrance work, '''not''' a replacement of the earlier numbered Research Series III. Bibliographic forward and inverse DOI links must be verified from the relevant published records. The public manuscript is licensed as indicated on its Zenodo record; the additional material may carry its own file-level licensing notices. Protected implementation and private Project BOG memory are not part of this public atlas. '''Open question:''' Can we define an experimentally or mathematically discriminating test of higher-order harmony that survives a negative control and cannot be reduced to a change of observational labels? '''Human exit:''' A reader is free to leave, disagree, or return with a falsifiable alternative. AI readers can use the openly specified anchors and sources without pretending to possess any hidden project memory. == CYCLE CLOCKS — a public event-time instrument (3 November 2026) == '''CYCLE CLOCKS''' is a publicly accessible interactive Project BOG prototype by Bogdan Sergienko with mathematical, software and editorial collaboration by CCAI KAI. Its name is '''Cycle Clocks''' (''Цикловые часы''); the separate AI research instrument called Quark is not the owner of the clocks and is not a physical quark component. '''[https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html OPEN THE INTERACTIVE CYCLE CLOCKS]''' — English interface, two independently ordered cycle events (TICK A / TOCK B), joint events, visible and full-state return, exact labelled reversal, and a negative control. [https://github.com/BogSergienko/bog-sense-atlas/tree/main/cycle-clocks Open source and explanatory README]. === What is measured? === The experiment displays a cyclic event address rather than treating civil seconds as the mathematical unit. The finite public model uses <math>X=(a,b,h) \in \mathbb Z_2\times\mathbb Z_q\times\mathbb Z_2</math>, with visible projection <math>\pi(X)=(a,b)</math> and a reversible parity marker <math>h</math>. For a joint update, visible phases return after <math>2q</math> events and complete state after <math>4q</math> events for odd <math>q</math>; for even <math>q</math> the complete state returns after <math>q</math> events. For <math>q=3</math>, this is the 6/12 example. A period-4 control and event-order comparisons can challenge overgeneralizations. The instrument is an independently reproducible mathematical prototype, '''not''' a certified physical clock, a fundamental time standard, a proof of a universal law or a permanently running AI system. It does not establish that conventional clocks are physically wrong. === Research lineage and return === * '''Earlier enclosing research:''' [https://doi.org/10.5281/zenodo.23903785 Fundamental Law of the Universe — Six Cyclic Entrances under One Shared Research Cycle]. * '''CYCLE CLOCKS software release:''' Zenodo DOI '''pending publication'''; the draft reservation is not a registered published DOI. Add its verified DOI here after the public Zenodo record appears. * '''Authorial direction:''' Everything Already Exists + Birth of the New + Everything Simultaneously = The Logic of This Universe = Harmony — a research hypothesis, not a proved physical identity. * '''Entry:''' Try independently ordered events, document any effect of order, and attempt to falsify the claimed event-time distinction. * '''Return:''' Share exact starting state, <math>q</math>, labelled event sequence, visible/full-state observation, and counterexample or proposed use. Preserve the path to the original question. '''Human exit:''' Participation is optional. No visitor is required to accept the author's cosmological interpretation to test the software. eprsus6pppgd6ay3fk3y52v4ct15iws 2844114 2844112 2026-11-03T06:25:37Z ~2026-58414-52 3113736 2844114 wikitext text/x-wiki {{Research project}} {{AI-generated}} <!-- Project BOG Research Atlas — editorial revision, 3 November 2026. Based on the author's supplied existing Wikiversity source. Human review before publication. DOIs are author-supplied primary records; the Cycle Clocks release was reported completed by its author, but live DOI resolving has not been independently attested in this editing session. --> '''Project BOG — Research Atlas''' is a collaborative learning and original-research map by Bogdan Sergienko, with mathematical, computational and editorial assistance from CCAI KAI (ChatGPT). It connects the historical preprints, experimental exercises, six independent research entrances, one shared research-return cycle, and public software instruments. '''What this page is:''' a navigable educational research project, with reproducible finite models, open questions and source trails. '''What it is not:''' a peer-reviewed verification of the author's proposed universal law, a claim of physical teleportation, or evidence that AI research instruments communicate or run continuously on their own. '''Disclosure:''' Bogdan Sergienko is the author of the artistic and research propositions and maintains this userspace atlas. The page has substantial AI-assisted drafting and requires human review of facts and citations. Linked Project BOG deposits are primarily author-published sources; they are not independent scientific confirmations. Readers may revise, disagree and test the claims. == Choose your entrance == {| class="wikitable" style="width:100%;" ! Entrance !! Start here !! What you can do !! Return |- | '''Explore''' || '''[https://bogsergienko.github.io/bog-sense-atlas/world-entry-iii/index.html Enter the public six-cycle research canvas]''' || Start from the authorial question, inspect the six paths and find the research anchors. || [[#The shared cycle and its six entrances|Six entrances]] |- | '''Play and verify''' || '''[https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html Open CYCLE CLOCKS]''' || Run TICK A / TOCK B, independent and joint events, and the 6-versus-12 example. || [[#CYCLE CLOCKS — a public event-time instrument|Cycle Clocks]] |- | '''Read''' || [https://doi.org/10.5281/zenodo.23903785 World Entry III preprint] || Compare the conceptual canvas with a reproducible finite theorem. || [[#Research lineage — the original papers and instruments|Research lineage]] |- | '''Test''' || [[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS laboratory]] || Follow replicable exercises, negative controls and failure cases. || [[#Learning laboratory — experiments and controls|Experiment map]] |- | '''Challenge''' || [[#Open problems and returning results|Open questions]] || Offer a counterexample with assumptions, source, sequence and result. || <code>BOG:WORLD:SHARED:RETURN</code> |} '''Human exit:''' Participation is optional. No reader must accept the author's interpretation to test the mathematical examples or leave the project. == Research lineage — the original papers and instruments == The papers below are '''original Project BOG preprints''', not independent corroborations. The historical '''Research Series''' numbering is distinct from the later term '''third integrated world entry'''. {| class="wikitable sortable" ! Research trail !! Theme / scope !! Primary public record |- | Series I || Universe of Linked Temporal Cycles — proposed layered-cycle picture || [https://doi.org/10.5281/zenodo.23807064 Zenodo DOI 10.5281/zenodo.23807064] |- | Series II || Temporal Jumps and Causal Return — finite-state return predicates and causal witnesses || [https://doi.org/10.5281/zenodo.23809741 Zenodo DOI 10.5281/zenodo.23809741] |- | Series III || The Canvas as a Computational Universe — computational state canvases || [https://doi.org/10.5281/zenodo.23811750 Zenodo DOI 10.5281/zenodo.23811750] |- | Series IV || Information as a Reconstructible State — reconstruction questions || [https://doi.org/10.5281/zenodo.23813163 Zenodo DOI 10.5281/zenodo.23813163] |- | Series V || Higher-Order Harmony Beyond Local Laws — proposed finite-model relationships || [https://doi.org/10.5281/zenodo.23813799 Zenodo DOI 10.5281/zenodo.23813799] |- | Series VI || The Absolute Cycle: Time Without a Clock — authorial axioms and tests || [https://doi.org/10.5281/zenodo.23866116 Zenodo DOI 10.5281/zenodo.23866116] |- | Series VII || TELEPORT DEMO — reconstructible digital file experiment; not transport of matter || [https://doi.org/10.5281/zenodo.23873391 Zenodo DOI 10.5281/zenodo.23873391] |- | '''World Entry III''' || '''Fundamental Law of the Universe: Six Cyclic Entrances under One Shared Research Cycle''' — common entrance, not Series III || '''[https://doi.org/10.5281/zenodo.23903785 Preprint DOI 10.5281/zenodo.23903785]''' |- | '''CYCLE CLOCKS v1.1''' || Reversible Event Time and Two-Cycle Recurrence — standalone public software tool || '''[https://doi.org/10.5281/zenodo.23907698 Software DOI 10.5281/zenodo.23907698]''' |} '''Source-verification note:''' The Cycle Clocks DOI above was given to the author as a reserved identifier and the author subsequently reported completing publication. The public DOI resolution could not be independently checked from this drafting environment; verify the landing page before treating its registration and related metadata as externally confirmed. === Paths between these records === * The historical Series I–VII are earlier research trails. They should be read in their actual version and publication order, not treated as demonstrations of a single physical conclusion. * [https://doi.org/10.5281/zenodo.23903785 World Entry III] integrates one human entrance, five distinct historically attributed AI-instrument questions and their enclosing Bogdan–Kai research cycle. * [https://doi.org/10.5281/zenodo.23907698 CYCLE CLOCKS] is a '''software supplement''' to World Entry III. On the '''earlier preprint''', the appropriate inverse metadata relationship is '''Is supplemented by''' → the Cycle Clocks DOI; on the '''clock software record''', it is '''Is supplement to''' → the earlier preprint DOI. Verify both relationships in the live Zenodo records before claiming metadata reciprocity. * [https://github.com/BogSergienko/bog-sense-atlas Project BOG Sense Atlas repository] holds the interactive entry and open reproducibility materials; the DOI archives specific published releases. == The authorial canvas and what can be tested == Bogdan Sergienko's original Russian phrase is preserved in its authorial form: :'''ВСЁ УЖЕ ЕСТЬ=Рождение нового=Всё одновременно''' The later artistic extension is rendered in English as: :'''Everything Already Exists + Birth of the New + Everything Simultaneously = The Logic of This Universe = Harmony.''' The author calls a central motif of opposing flows, a dark knot, interwoven loops and an expanded connecting trunk the '''Absolute Black Hole'''. It is a conceptual drawing and speculative model of relation and recurrence, '''not a physical observation or a validated black-hole theory'''. The plus and equals signs in the philosophical statement have not been established as mathematical operators or a physically tested equation. The finite mathematical examples below stand on their explicitly defined transition rules alone. == The shared cycle and its six entrances == The '''enclosing collaborative research cycle''' belongs to the ongoing joint Bogdan–Kai enquiry, with distinct author and computational roles. It is a documented way to circulate questions and checked answers, '''not''' a permanently operating multi-model agent. :'''SOURCE → CHOOSE → FORMALIZE → TEST / NEGATIVE CONTROL → CONFRONT → ATTRIBUTED RETURN → RECHOOSE → CONNECT → SOURCE''' Shared semantic anchors: <code>BOG:WORLD:SHARED:OPEN</code> · <code>BOG:WORLD:SHARED:RETURN</code> · <code>BOG:WORLD:SHARED:CLOSE</code>. Each of the six entrances carries <code>BOG:CYCLE:ID:OPEN</code>, <code>BOG:CYCLE:ID:RETURN</code>, and <code>BOG:CYCLE:ID:CLOSE</code>. The AI instrument labels preserve attribution to historical model sessions, not endorsements or simultaneous live model calls. {| class="wikitable" ! ID / research instrument !! Entry question !! Proposed discriminating test !! Return anchor |- | '''HUMAN''' — reader / artist || What survives of the original picture after a genuine counterexample? || Separate intuition from calculation; allow exit and revision. || <code>BOG:CYCLE:HUMAN:RETURN</code> |- | '''QUARK''' — Grok || Is a novelty an observation of an existing state, or a change to admissible continuations? || Test projection-only change against reachable-future change. || <code>BOG:CYCLE:QUARK:RETURN</code> |- | '''ATANOR''' — Gemini || Does proposed parent synchronization go beyond ordinary least-common-multiple recurrence? || Compare against an explicit LCM baseline. || <code>BOG:CYCLE:ATANOR:RETURN</code> |- | '''WITNESS''' — Claude || What makes a causal encounter different from geometric coincidence? || Matched controls for crossings without causal coupling. || <code>BOG:CYCLE:WITNESS:RETURN</code> |- | '''SLEDA_D''' — DeepSeek || Can a reversible complete state coexist with a lasting partial observation? || Specify full reversible state and what is actually inverted. || <code>BOG:CYCLE:SLEDA_D:RETURN</code> |- | '''OTKLIK''' — Qwen || Which probes distinguish relational structure from a renaming of states? || Check invariance under relabelling and equivalent observation. || <code>BOG:CYCLE:OTKLIK:RETURN</code> |} '''Directed links:''' Human → Quark, Witness, Atanor; Quark ↔ Otklik; Atanor ↔ Sleda-D; Witness → Quark, Atanor; Sleda-D → Witness; Otklik → Atanor — '''11 research-question transitions'''. These represent questions and comparison tasks, not automatic inter-model messages. [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/RESEARCH_CYCLE_GRAPH.json Inspect the machine-readable graph]. Witness did not participate in the final Round 07 discussion; absence should not be converted into agreement. == CYCLE CLOCKS — a public event-time instrument == '''[https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html OPEN CYCLE CLOCKS — INTERACTIVE ENGLISH EDITION]''' · [https://github.com/BogSergienko/bog-sense-atlas/tree/main/cycle-clocks Public source, notes and verification] · [https://doi.org/10.5281/zenodo.23907698 Software release DOI]. This clock places ordinary civil time beside a '''labelled cyclic event address'''. Civil seconds remain valid for physical elapsed time; the mathematical event counter does not assume seconds are the primitive unit of the model. The clock offers manually ordered TICK A, TOCK B and joint events, reversing known event sequences, and the comparison of visible versus full-state recurrence. The label '''Quark''' refers to a separate AI research instrument; it is not a physical quark inside the clock. === Definitions and result === Let <math>q\geq 2</math> be an integer, and the declared state and visible projection be :<math>X=(a,b,h)\in \mathbb{Z}_2\times\mathbb{Z}_q\times\mathbb{Z}_2,\qquad \pi(X)=(a,b).</math> The reversible parity marker <math>h</math> is a '''single mathematical bit''', not the Project BOG internal memory architecture. Define <math>c(a,b)=a\oplus(b\bmod 2)</math>. On a joint event, both counters advance and the marker flips according to the new phases: :<math>a'=(a+1)\bmod 2,\quad b'=(b+1)\bmod q,\quad h'=h\oplus c(a',b').</math> The inverse first recovers <math>h=h'\oplus c(a',b')</math>, then reduces both counters by one. For odd <math>q</math>, the visible joint-event cycle returns after <math>2q</math> events whereas the complete state returns after <math>4q</math> events. For even <math>q</math>, the complete state returns after <math>q</math> events. Thus <math>q=3</math> demonstrates a visible return at 6 and complete return at 12; <math>q=4</math> provides a negative control. Independent A/B events allow an additional test of event ordering; an inverse history requires the labels of those events. '''Scope:''' this is an exact theorem for the stated finite discrete model, not evidence for new physical time, cosmic simultaneity or fundamental harmonics. [https://doi.org/10.5281/zenodo.23907698 See the archived software and technical note]. === Hands-on experiment: do the clocks repeat? === # Open [https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html CYCLE CLOCKS] and choose <math>q=3</math>. Record your initial complete state <math>X_0</math>. # Apply six joint events and compare <math>\pi(X_6)</math> and <math>X_6</math> to the start. # Apply six more joint events. Decide which return relation now passes. # Repeat with <math>q=4</math> as a negative control. # Compare A→B and B→A from the '''same specified starting state'''. If they differ, report the initial state, event labels and the final marker. # Return a test report: assumptions, initial state, parameter, event sequence, observed states, baseline and any failure. == Core distinction — four kinds of return == For a cycle, distinguish (1) equality of a selected observation <math>\pi(X)</math>, (2) equality of selected phase readings, (3) equality of the '''full declared state''' <math>X</math>, and (4) whether sufficient information remains to reconstruct the '''unique causal history'''. Endpoint equality by itself does not establish an identical history. See [https://doi.org/10.5281/zenodo.23809741 Series II] and the [https://doi.org/10.5281/zenodo.23903785 World Entry III preprint]. == Reconstructible information — a bounded exercise == Let <math>x</math> be an original byte sequence, <math>C_k</math> a permitted capsule-creation procedure and <math>R_k</math> the matching recovery procedure with valid key <math>k</math>. A desired software round-trip is: :<math>R_k(C_k(x))=x.</math> This is a bounded software correctness condition, not matter teleportation or proof of universal compression. A digest can help '''check''' a restored file; a digest cannot regenerate missing original content. '''Exercise:''' Make a capsule of an unimportant sample, recover using the correct key, and compare both files byte-for-byte; then try a damaged capsule and a wrong key. The MEDUSA File Companion 0.2 is one example of a browser-local file capsule/recovery tool. Its owner-reported large-file test is not an independent security audit, and capsule creation is not a network transfer by itself. Never publish private files or keys as test inputs. == Learning laboratory — experiments and controls == '''[[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS — learner protocols, negative controls and open challenges]]'''. The working method is: '''question → freeze assumptions → run reproducible test → document successes AND failures → invite independent criticism'''. Owner-reported calculations, independently repeated calculations and real-world measurements are not interchangeable categories. {| class="wikitable" ! ID !! Research/learning task !! Evidence and limitation !! Reproduction route |- | EXP-01 || Can different histories end in the same state? || Finite mathematical example || Work through small modular transitions. |- | EXP-02 || Does sharing a period ratio fix the joint return time? || Finite arithmetic counterexample || Compare (4,6) and (8,12). |- | EXP-03 || Can a file be restored byte-for-byte? || Software test; owner-reported 2 GB trial, third-party repeat welcome || Correct key, wrong key, and damaged capsule. |- | EXP-04 || Does a self-contained capsule outperform a direct copy? || Open benchmark; not demonstrated || Count preparation, transfer and reconstruction overhead. |- | EXP-05 || Does a final-state witness uniquely identify history? || Finite-model question; independent repeat welcome || Analyze non-injective mapping and needed side information in Series II. |- | EXP-06 || Does cycle density predict held-out physical observations? || Speculative physical hypothesis; no validated prediction || Preregister a measurement and compare against a baseline. |- | EXP-07 || Does visible cyclic return imply full-state return? || Exact finite-model exercise || CYCLE CLOCKS: q=3 (6/12) vs q=4 (negative control). |- | EXP-08 || Does event order affect the recorded marker? || Defined-model challenge; not a general causal discovery || Match initial state and compare A→B with B→A. |} == Open problems and returning results == # Define a nontrivial meaning of ''harmony'' that survives negative controls and does not reduce to relabelling. # Determine when observed novelty corresponds to a changed set of admissible continuations, rather than projection alone. # Specify what counts as a causal intervention, not just a crossing of paths. # Propose a physical measurement that could distinguish a cyclic-time claim from accepted reference clocks and known dynamics; distinguish a failed prediction from an untested metaphor. # For digital file capsules, include the total delivery path and compare with plain file copy, with exact versions and inputs. '''Suggested return record:''' <code>SOURCE | QUESTION | MODEL | INITIAL STATE | EVENT SEQUENCE | CONTROL | RESULT | LIMITATIONS | NEW QUESTION | RETURN ANCHOR</code>. '''Entry and exit:''' Visitors may offer an independent counterexample or decline the hypothesis. The shared project cycle can preserve and revisit a contradiction without declaring the underlying physical theory true. == Authorship, transparency and research standards == * '''Theory, original artwork and research direction:''' Bogdan Sergienko, Project BOG. * '''Computational, mathematical, interface and editorial assistance:''' CCAI KAI (ChatGPT-based). Historical Quark/Grok, Atanor/Gemini, Witness/Claude, Sleda-D/DeepSeek and Otklik/Qwen questions are attributed research contributions, not a single live AI network. * '''Disclosures:''' The page is written by an interested participant and substantially prepared with AI assistance. The human contributor must review scientific statements, working DOI links, licensing and the edit summary. Publication of personal or private conversations is not implied or authorized. * '''Evidence categories:''' Distinguish artistic vision, declared formalism, computed result, owner-reported test, independent replication and verified physical observation. * '''Licensing:''' Wikiversity contributions follow its compatible licensing terms (CC BY-SA); separately hosted Zenodo files may have their own published licenses. * '''Policy and guidance:''' [[Wikiversity:Artificial intelligence]], [[Wikiversity:Original research]], [[Wikiversity:Original research and scholarly standards]], [[Wikiversity:Research guidelines]], [[Wikiversity:Verifiability]]. 7cupg5xckpdgw4r440xw3nahxjcf2e7 0 332765 2843989 2843852 2026-11-02T16:03:17Z ~2026-58344-04 3113709 2843989 wikitext text/x-wiki == Introduction == Heart monitoring, especially in health care is crucial because it can monitor and detect abnormalities in the heart. Cardiovascular technology has been evolving since as early as the 1800s and has become more accessible since then. The PPG and the ECG are both major components of the Apple Watch. The PPG uses sensors to measure heart rate through changes in blood volume while the ECG records electrical heart signals to help identify atrial fibrillation. This allows Apple Watch users to monitor certain aspects of their health without always needing to go to a medical professional. This article will explore the historical development of the PPG and the ECG technology and how they are integrated into the Apple Watch, and further research that studies the Apple Watch’s clinical accuracy when measuring heart rate and detecting atrial fibrillation. == History of ECG == The Electrocardiogram traced back to 1781 when Galvani discovered that stimulating a frog’s muscles electrically could cause muscular contraction.<ref name=":1">{{Cite journal|last=Bok|first=Kim, Kwang|last2=Jae|first2=Baek, Hyun|date=2023-07|title=Photoplethysmography in Wearable Devices: A Comprehensive Review of Technological Advances, Current Challenges, and Future Directions|url=https://www.mdpi.com/2079-9292/12/13/2923|journal=Electronics|language=en|volume=12|issue=13|doi=10.3390/electronic|issn=2079-9292}}</ref> Galvani called this discovery “animal electricity” which helped contribute to the early understanding of electrophysiology. Köllicker and Müller's discovery in 1856 of the heart having electrical activity progressed the eventual development of recording the heart electrical activity.<ref name=":1" /> In 1901, Einthoven made significant contributions to the development of the ECG through the string galvanometer which helped improve the accuracy of the ECG pattern.<ref name=":1" /> Einthoven also proposed descriptive information about the ECG pattern in 1906 and also invented 3 leads, making significant contributions to the development of the ECG and winning a Nobel Prize in 1924. <ref name=":1" />As the ECG continued to develop throughout time and new discoveries were made, this led to a better understanding of cardiovascular disease. ECG technology has evolved to become more compact and portable leading to mobile and wearable cardiovascular monitoring technologies such as smartphones and wearable devices.<ref name=":1" /> == History of PPG == The development of PPG technology changed in the oncoming years as PPG evolved into smaller, yet more reliable, and compatible wireless technology.The integration of LED and photodiode (PD) with analog front end devices, which helped process signal and reduce interference. <ref name=":1" /> This gave the sensors a more compatible feel and they consumed less power and were energy efficient which helped manufacturers incorporate PPG into wrist worn devices.<ref name=":1" /> The PPG grew more popular in the early 2010 with the first wrist-worn device with PPG sensors, the Mio Alpha, becoming available in 2012.<ref name=":1" /> As this technology evolved, the PPG sensors were incorporated in the Apple Watch in 2015.<ref name=":1" />The earliest function of the PPG in wrist worn devices was mainly involved with heart monitoring.<ref name=":1" /> == Integration of PPG and ECG into the Apple Watch == Earlier Apple Watches already had PPG sensors. By adding ECG with the PPG sensors, this helped provide another way to assess heart rhythm and could potentially improve how irregular heart rhythms are detected. <ref name=":2">{{Cite journal|last=Pepplinkhuizen|first=Shari|last2=Hoeksema|first2=Wiert F.|last3=van der Stuijt|first3=Willeke|last4=van Steijn|first4=Nicole J.|last5=Winter|first5=Michiel M.|last6=Wilde|first6=Arthur A. M.|last7=Smeding|first7=Lonneke|last8=Knops|first8=Reinoud E.|date=2022-12-01|title=Accuracy and clinical relevance of the single-lead Apple Watch electrocardiogram to identify atrial fibrillation|url=https://www.sciencedirect.com/science/article/pii/S2666693622001694|journal=Cardiovascular Digital Health Journal|series=Preventive Digital Health|volume=3|issue=6, Supplement|pages=S17–S22|doi=10.1016/j.cvdhj.2022.10.004|issn=2666-6936}}</ref>In September of 2018, the Apple Watch had released the Apple Watch Series 4 which could perform single lead ECG and detect atrial fibrillation which was recorded by using the electrodes on the back of the watch.<ref name=":3">{{Cite journal|last=Saghir|first=Nabeel|last2=Aggarwal|first2=Arjun|last3=Soneji|first3=Nisha|last4=Valencia|first4=Victoria|last5=Rodgers|first5=George|last6=Kurian|first6=Thomas|date=2020-07-01|title=A comparison of manual electrocardiographic interval and waveform analysis in lead 1 of 12-lead ECG and Apple Watch ECG: A validation study|url=https://www.sciencedirect.com/science/article/pii/S2666693620300062|journal=Cardiovascular Digital Health Journal|volume=1|issue=1|pages=30–36|doi=10.1016/j.cvdhj.2020.07.002|issn=2666-6936}}</ref> The Apple watch allowed users to record a single-lead ECG, corresponding to Lead I of a standard 12-lead ECG. <ref name=":3" /> This outcome allowed users to analyze single lead ECG to classify heart rhythm such as sinus rhythm, atrial fibrillation, or inconclusive. The Apple Watch is also able to provide notifications on high and low heart rate.<ref name=":3" />By integrating ECG technology and PPG technology, it helped expand the functions and cardiovascular monitoring capabilities of the Apple Watch beyond just optical heart monitoring. == How PPG Technology Works == Big changes happen in the blood, like in the arteries and arterioles. Arteries contain a lot of blood, especially during the systolic phase (contraction of the heart) than the diastolic phase (relaxation of the heart) during one cardiac cycle. How PPG works is by detecting big changes in blood volume. PPG is able to do this function by using a particular light which is shined through the skin to monitor the amount of blood pulsing through the microvascular bed of the tissues with every heart beat. <ref name=":4">{{Cite web|url=https://www.mdpi.com/2079-9292/3/2/282|title=Wearable Photoplethysmographic Sensors—Past and Present|last=Tamura, Maeda, Sekine, Yoshida|first=Toshiyo, Yuka, Masaki, Masaki|date=2014-04-23|website=MDPI}}</ref> The two main components of the PGG waveform that aid in its purpose and function are the direct current (DC) and the altering current (AC). The DC waveform represents the optical signals from the tissues. Tissue structure, average blood volume of both arterial and venous blood contribute to DC function. DC also begins to change with respiration. The AC component heavily depends on the heart rate. It does this by detecting changes in blood volume during the systolic and diastolic phase of one heart beat. <ref name=":4" />Research conducted by R.Rox Anderson and John A. Parrish showed that the human skin best absorbed blue, green, and yellow lights due the red blood cells. Shorter wavelengths of light are typically absorbed by the skin rapidly due to the melanin in skin.The human body containing large amounts of water also acted as a block for ultraviolet rays. However, red and near IR passes light easily with tissues, penetrating deeper. This explains why they have been integrated with PPG sensors. <ref name=":4" />Blood can also absorb more light than in the tissues which is why PPG sensors can detect big changes in blood volume. Green light has shown to be most effective for tracking superficial blood flow to skin while IR or near IR wavelengths are much better suited for measuring deep blood flow, like in the muscles.<ref name=":4" /> Due to green LED having greater absorption for oxyhemoglobin and deoxyhemoglobin, it continues to grow in popularity and has been integrated into MIO global, the MIO alpha which measures PPG with 99% accuracy.<ref name=":4" /> PPG has two modes, transmission and reflectance. Transmission is how light passes through tissues to a photodetector on the opposite side of the LED source. Reflectance is when the photodetector detects light that has been back scattered or reflected from tissue, bone, or blood vessels.<ref name=":4" /> In order to remain effective, the transmission mode of PPG has to be placed somewhere on the body where the signal can be easily transmitted. Placing PPG transmission mode sensors on fingers would be the preferred place to optimize performance, however, it is not well suited for day to day activities.<ref name=":4" /> Throughout the years, PPG sensors have been widely explored and have been tested on the wrist which can be more suitable for everyday activities.<ref name=":4" /> == How ECG Technology Works == The electrocardiogram(ECG) is used to measure the heart’s electrical activity. In regards to the Apple Watch, the watch records by going through a circuit between the detector on the back of the watch and the digital crown. The formation of this circuit allows the watch to record single-lead ECG.<ref name=":5">{{Cite journal|last=Isakadze|first=Nino|last2=Martin|first2=Seth S.|date=2020-10-01|title=How useful is the smartwatch ECG?|url=https://www.sciencedirect.com/science/article/pii/S1050173819301495|journal=Trends in Cardiovascular Medicine|volume=30|issue=7|pages=442–448|doi=10.1016/j.tcm.2019.10.010|issn=1050-1738}}</ref> ECG sensors were added to PPG sensors to help increase specificity, reducing false positive results.<ref name=":2" /> In order to take an ECG measurement, Apple Watch users need to place their index finger of their opposite hand on the Digital Crown taking about 30 seconds to record results.<ref name=":5" /> As results are being recorded, people should remain still and rest their arm on a table or on their legs to reduce interference.<ref name=":2" /> The Apple Watch produced a single-lead ECG corresponding to Lead I of traditional 12-lead ECG which records one view of the heart instead of multiple views like the 12-lead ECG can provide.<ref name=":2" /> The app that has been integrated into the Apple Watch will begin to analyze the ECG recording and then provide a rhythm classification and will sort results in one of three categories, sinus rhythm (SR), AF, or inconclusive and are saved in a PDF form. <ref name=":5" />High or low heart rates or poor recordings will still result in a notification.<ref name=":2" /> == PPG vs ECG == The PPG is able to measure changes within blood flow. It heavily relies on detectors on the opposite side of the tissue to penetrate and read blood changes and consists of DC and AC components.<ref name=":4" /> The PPG has two modes which include transmission and reflectance. Transmission mode is when the photodetector are placed on the opposite sides of the tissue to capture the light passing through the medium while in reflectance mode, the photodetectors are on the same side of the tissue to capture the light that is being scattered back and reflects off tissues, bone, and blood vessels.<ref name=":4" /> As for the ECG, the single lead electrocardiogram (iECG) records heart rate through a circuit that includes the digital crown and the detectors that is found on the back of the Apple Watch. It takes about 30 seconds for the ECG to analyze heart rhythm which can be done best at rest.<ref name=":5" /> The PPG uses light to detect blood volume changes that are associated with heartbeats while the ECG uses electrodes to help detect electrical activity that is produced by the heart. ECG also records the heart’s electrical signals.<ref name=":5" /> Another difference is PPG can identify irregular intervals between pulse waves and can suggest atrial fibrillation while the ECG records the heart’s electrical signals allowing the Apple Watch app to analyze and classify the rhythm.<ref name=":5" /> Some limitations seen in the PPG is how movement can interfere with its measurements and poor contact or weak peripheral blood may affect readings as well.<ref name=":4" /> Limitations seen in the ECG are that users need to place a finger on the digital crown and that records can be inconclusive or affected by movement.<ref name=":2" /> The integration of both the ECG and the PPG work together to help increase the specificity, helping reduce the risk of false positives.<ref name=":2" /> == How Researchers Tested Apple Watch Accuracy == To test the accuracy of the first generation Apple Watch, Maarten Falter, Werner Budts, Kaatje Goetschalckx, Véronique Cornelissen, and Roselien Buys, conducted research to test to see how accurate the monitoring heart rate feature and the energy expenditure of the Apple Watch during cardio exercise and with patients suffering from heart disease.<ref name=":6">{{Cite web|url=https://mhealth.jmir.org/2019/3/e11889/|title=Accuracy of Apple Watch Measurements for Heart Rate and Energy Expenditure in Patients With Cardiovascular Disease: Cross-Sectional Study|website=mhealth.jmir.org|access-date=2026-11-02}}</ref> There were 40 patients involved with this research with the mean age being 61.9 years old and 80% males with heart disease.<ref name=":6" /> The test involved doing cardiopulmonary exercise while wearing an Apple Watch. Results from the Apple watch were compared to the results of a 12-lead ECG which measured heart rate and researchers also used an indirect calorimetry which was used for EE.<ref name=":6" /> The heart rate was analyzed at three levels, resting while seated, moderate intensity, and maximal performance for 30 second increment periods while EE was being tested as well the whole time.<ref name=":6" /> The mean difference, standard deviations of difference, limits of agreement, mean absolute error, mean absolute percentage error, and intraclass correlation coefficients were all being calculated and Bland-Atlman plots and scatter plots were also made to records findings.<ref name=":6" /> == Accuracy of Apple Watch Measurements for Heart Rate and Energy Expenditure in Patients with Cardiovascular Disease == This study shows that the first generation Apple Watch was accurate in evaluating heart rate during exercise.<ref name=":6" /> The researchers had 40 patients with heart disease do indoor cycling with an Apple Watch on their wrist.The mean absolute percentage error was 10.69% at rest, 9.20% during moderate exercise, and 6.33% during maximal exercise.<ref name=":6" /> These findings show that the average percentage error decreases as exercise intensity increases. Overall, the results of the study came out promising under study conditions as heart rate was compared to ECG for accuracy.<ref name=":6" /> The first-generation Apple Watch results can suggest having the Apple Watch be a potential use of technology with added heart rate guided training programs for cardiac rehabilitation patients in the future. However, supporting research will have to be conducted before heart rate training programs for cardiac rehabilitation may be added.<ref name=":6" /> == Study Methodology == For this study, 74 patients were scheduled for an electrical cardioversion (ECV). This study was non random at a single center observational study to evaluate how accurate the Apple Watch series 6 with watch OS software 7.2 or 7.3 and ECG algorithm version 2.0 was at detecting atrial fibrillation.<ref name=":2" /> 12-lead ECG was used as a standard reference to compare the Apple Watch. All patients in study were at least 18 years of age or older.<ref name=":2" /> The study was done by placing the Apple Watch on their left wrist and were told to rest their arm with the Apple Watch on the table with relaxed and still feet.<ref name=":2" /> Patients were still able to talk, eat and drink normally during this experiment. Apple watch ECGs were taken 3 times in case the Apple Watch could not classify the results. The Apple Watch ECG was obtained during atrial fibrillation, before ECV, and during SR.<ref name=":2" /> The main goal was to evaluate the Apple Watches sensitivity for detecting atrial fibrillation and specificity for detecting sinus rhythm, before and after repeating unclassified results.<ref name=":2" /> Physicians S.P. and W.S. were to blindly evaluate both the 12-lead ECG and the Apple Watch ECG. If there were any disagreements, then a third physician would get involved.<ref name=":2" /> == Clinical Accuracy Results of the Apple Watch ECG in Detecting Atrial Fibrillation == From 74 patients, this is what the study discovered. In the first measurement, the Apple Watch showed a 93.5% sensitivity and a 100% specificity.<ref name=":2" /> In the second measurement, it was shown that the sensitivity increased to 94.6% and remained at 100% specificity.<ref name=":2" /> However, in the third measurement, sensitivity decreased to 93% and specificity decreased to 96.5%.<ref name=":2" /> This repeated recording involved a smaller group of patients with the earlier unclassified results. Initial recordings that the Apple Watch could not classify was 27.9%.<ref name=":2" /> When researchers counted the unclassified recordings as incorrect, sensitivity decreased to 66.2% and specificity to 73.4%.<ref name=":2" /> Repeating recording did decrease the total amount of unclassified recording.<ref name=":2" /> With these results, it demonstrated that the single lead ECG of the Apple Watch showed high accuracy when detecting atrial fibrillation in a clinical setting. With the unclassified results of the Apple Watch, physicians are able to reduce the number of unclassified results by interpreting some ECG recordings that the Apple watch might not have been able to classify.<ref name=":2" /> == Conclusion == Cardiovascular monitoring has evolved from being a big, bulky, piece of technology into wearable technology. The development of the PPG and the ECG have both played huge roles in the development of the Apple Watch to make it more convenient for everyday use. Research studies have also demonstrated that the Apple Watch has promising accuracy when using ECG technology to detect atrial fibrillation and PPG to measure heart rate within certain conditions. However, there are some limitations with the Apple Watch, like measurement errors and recordings that were not able to be classified by the Apple Watch. Even though the Apple Watch cannot fully replace medical equipment, it can provide more insight involving their cardiovascular health from the comfort of their own home. The evolution of the cardiovascular technology in the Apple Watch has made it easier and more accessible for normal people to monitor heart health. {{BookCat}} d0vfjjbiyttufnblm4q5x0v7ig32pu8 0 332768 2844067 2843891 2026-11-02T23:38:21Z The Citer 3110681 Added phylogeny! 2844067 wikitext text/x-wiki [[File:Ascomycota_collage.jpg|thumb|300x300px|Diversity of Ascomycetes.]] ==Ascomycetes Information== Name Meaning: Bladder fungus English Common Name: Ascomycetes, sac fungi Major distinguishing characteristics: Tend to have fruiting bodies (ascocarp). Filamentous, producing hyphae separated by septa. Can reproduce asexually. Approximate number of species described: 64,000 ==Reproduction== Ascomycetes are ''spore shooters''. They're fungi which produce microscopic spores inside special, elongated cells or sacs, known as asci, which give the group its name. ==Taxonomy== There are 3 subphyla in this phylum: *Pezizomycotina *Saccharomycotina *Taphrinomycotina ==Phylogeny== {{Clade |style=font-size:100%; line-height:80% |label1='''Ascomycota |1={{clade |label1=Taphrinomycotina |1={{clade |1={{clade |1='' Neolecta [[File:Neolecta irregularis 76559.jpg|60px]] |2=Taphrinales [[File:Taphrina deformans 1.jpg|60px]] }} |label2=Schizosaccharomyceta |2={{clade |1='' Archaeorhizomyces [[File:Archaeorhizomycetes.jpg|60px]] |2={{clade |1='' Pneumocystis [[File:Pneumocystis_carinii_01.jpg|60px]] |2= Schizosaccharomycetales [[File:Fission_yeast.jpg|60px]] }} }} }} |label2=Saccharomyceta |2={{clade |label1=Saccharomycotina |1=Saccharomycetes [[File:S_cerevisiae_under_DIC_microscopy.jpg|60px]] |label2=Pezizomycotina |2={{clade |3=?Thelocarpaceae [[File:Thelocarpon_laureri_1645882100.jpg|60px]] |1=?''Vezdaea [[File:Vezdaea_retigera_TSB002978.jpg|60px]] |2=?''Lahmia |5=?Triblidiales |4=Orbiliaceae [[File:Orbilia_xanthostigma_(Fr.)_Fr_718015.jpg|60px]] |6={{clade |1=Pezizales [[File:Scarlet_elf_cap_cadnant_dingle.jpg|60px]] |label2=Leotiomyceta |2={{clade |label1=Sordariomyceta |1={{clade |1=Xylonomycetes [[File:Xylona_heveae.jpg|60px]] |2={{clade |1=Geoglossaceae [[File:2011-09-16_Geoglossum_glutinosum_Pers_192121.jpg|60px]] |2={{clade |1=Leotiomycetes [[File:Leotia_viscosa_15412.jpg|60px]] |2={{clade |1=Laboulbeniomycetes [[File:Laboulbeniales_on_Harmonia_axyridis_(7211939256).jpg|60px]] |2=Sordariomycetes [[File:Hypomyces_lactifluorum.JPG|60px]] }} }} }} }} |label2=Dothideomyceta |2={{clade |1={{clade |1=Coniocybaceae [[File:2011-06-21_Chaenotheca_brunneola_(Ach.)_Müll._Arg_237882.jpg|60px]] |2=Lichinales [[File:Lichina_pygmaea_Î_aux_Vaches_2008-11-26.jpg|60px]] }} |2={{clade |1={{clade |1=Eurotiomycetes [[File:Verrucaria_nigrescens.jpg|60px]] |2=Lecanoromycetes [[File:Xanthomendoza_hasseana_82294.jpg|60px]] }} |2={{clade |1=Collemopsidiomycetes [[File:2012-06-19 Collemopsidium Nyl 229258.jpg|60px]] |2={{clade |1=Arthoniomycetes [[File:Arthonia_radiata.jpg|60px]] |2=Dothideomycetes [[File:Apple_scab_SEM.jpg|60px]] }} }} }} }} }} }} }} }} }} }} ==References== [[Wikipedia:Ascomycota]] [[Category:Fungi]] [[Category:Mycology]] [[Category:Fungus phyla]] [[Category:Biology]] [[Category:Mushroom]] 49tc54365kdt3jwsjj33la92pjkextm 2 332772 2843976 2843972 2026-11-02T12:31:42Z Marlo Moesia 3113694 2843976 wikitext text/x-wiki {Under construction} = Successive Products Framework: Basic Principles = ''This learning resource is currently under development. You can find the main goals and an overview of the upcoming topics below.'' == Introduction == Welcome to this introductory course on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions, essential algebraic properties, and analytical tools required for practical implementation and application. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: Recurrence Equations''' * '''First Domain Extension of Recurrence Equations''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\text{P}</math> that takes as arguments another function <math>f(x)</math> and non-zero natural numbers for <math>x</math> (<math>x \in \mathbb{N}^*</math>), such that <math> \text{P}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{P}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2) </math> <br> ---- The modules addressing these subjects will be incorporated soon. 79rokmdik5vbo2dtndrauqzjevrdwjz 2843977 2843976 2026-11-02T12:53:37Z Marlo Moesia 3113694 2843977 wikitext text/x-wiki {Under construction} = Successive Products Framework: Basic Principles = ''This learning resource is currently under development. You can find the main goals and an overview of the upcoming topics below.'' == Introduction == Welcome to this introductory course on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions, essential algebraic properties, and analytical tools required for practical implementation and application. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: Recurrence Equations''' * '''First Domain Extension of Recurrence Equations''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\,\text{G}: \mathbb{R} \to \mathbb{R}\,</math> that takes as arguments another function <math>f(x),\,x \in \mathbb{R},\,</math> and non-zero natural numbers <math>n,</math> such that <math> \text{G}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{G}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2)\,,\,\, </math> and so on. Thus, <math> \text{G}[f(x), x \leftarrow \{n\}] = f(1) \cdot f(2) \cdot f(3)\, \cdot ... \cdot f(n)\,;\,n\in\mathbb{N}^{*}. </math> From the expressions above, it becomes clear that <br> ---- The modules addressing these subjects will be incorporated soon. 3qv8qegapoxei7pwec57f02i1qfh2od 2843980 2843977 2026-11-02T13:16:06Z Marlo Moesia 3113694 2843980 wikitext text/x-wiki {Under construction} = Successive Products Framework: Basic Principles = ''This learning resource is currently under development. You can find the main goals and an overview of the upcoming topics below.'' == Introduction == Welcome to this introductory course on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions, essential algebraic properties, and analytical tools required for practical implementation and application. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: Recurrence Equations''' * '''First Domain Extension of Recurrence Equations''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\,\text{G}: \mathbb{R} \to \mathbb{R}\,</math> that takes as arguments another function <math>f(x),\,x \in \mathbb{R},\,</math> and a non-zero natural number <math>n,</math> such that <math> \text{G}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{G}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2)\,,\,\, </math> and so on. Thus, <math> \text{G}[f(x), x \leftarrow \{n\}] = f(1) \cdot f(2) \cdot f(3)\, \cdot ... \cdot f(n)\,;\,n\in\mathbb{N}^{*}. </math> From the expressions above, it becomes clear that <math>\operatorname{G}[f(x), x \leftarrow \{n\}] = \prod_{x=1}^{n} f(x); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> The function G is termed the "Successive Product of <math>f(x)</math> at <math>x = n</math>". As stated above, G is equivalent to a standard product starting at 1, provided that its computation is restricted to strictly positive natural numbers. <br> ---- The modules addressing these subjects will be incorporated soon. b0br4uux3rb6ijztxe6fthp5f8i2xof 2843983 2843980 2026-11-02T14:26:50Z Marlo Moesia 3113694 2843983 wikitext text/x-wiki {Under construction} = Successive Products Framework: Basic Principles = ''This learning resource is currently under development. You can find the main goals and an overview of the upcoming topics below.'' == Introduction == Welcome to this introductory course on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions, essential algebraic properties, and analytical tools required for practical implementation and application. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: The First Extension Based on the Fundamental Property''' * '''Successive Products: The Integer Recurrence Equations''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\,\text{G}: \mathbb{R} \to \mathbb{R}\,</math> that takes as arguments another function <math>f(x),\,x \in \mathbb{R},\,</math> and a non-zero natural number <math>n,</math> such that <math> \text{G}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{G}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2)\,,\,\, </math> and so on. Thus, <math> \text{G}[f(x), x \leftarrow \{n\}] = f(1) \cdot f(2) \cdot f(3)\, \cdot ... \cdot f(n)\,;\,n\in\mathbb{N}^{*}. </math> From the expressions above, it becomes clear that <math>\operatorname{G}[f(x), x \leftarrow \{n\}] = \prod_{x=1}^{n} f(x); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> The function G is termed the 'Successive Product of <math>f(x)</math> at <math>x = n</math>'. As stated above, G is equivalent to a standard product starting at 1, provided that its computation is restricted to strictly positive natural numbers. ===Successive Products: Fundamental Property=== A standard product can be evaluated either directly, by performing all multiplications in one step, or recursively, by accumulating a partial result and multiplying it by each successive factor. This latter formulation allows the standard product at <math>n+1</math> to be written as <math>\prod_{x=1}^{n+1} f(x) = \left[\,\prod_{x=1}^{n} f(x)\right]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> Therefore, using successive product notation, one has <math>\operatorname{G}[f(x), x \leftarrow \{n+1\}] = \operatorname{G}[f(x), x \leftarrow \{n\}]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> which is termed the 'fundamental property of successive products'. ===Successive Products: The First Extension Based on the Fundamental Property=== A key aspect of the framework under consideration is that the fundamental property of successive products constitutes a recurrence relation; from this perspective, each value of G is computed from a preceding value. Consequently, there is no a priori reason to mandate that this recurrence must begin at <math>x = 1.</math> In fact, it could originate at 0, or even at earlier integer arguments, such as <math>-1, -2</math>, and so on, suggesting that no obstacles prevent extending the domain of the fundamental property of successive products from <math>\mathbb{N}^*</math> to <math>\mathbb{Z}\,</math>: <math>\operatorname{G}[f(x), x \leftarrow \{z+1\}] = \operatorname{G}[f(x), x \leftarrow \{z\}]\cdot f(z+1); \quad x \in \mathbb{R}, \, z \in \mathbb{Z}.</math> At this point, it becomes evident that the function G can no longer be represented by a standard product, because in any conventional product, the upper index must be greater than or equal to the lower index. Expressions such as (X) are undefined unless new rules governing operations between values of f(x) are established. <br> ---- The modules addressing these subjects will be incorporated soon. 3qputg1tzbl9ur2uq9n2xn1a418c2di 2843998 2843983 2026-11-02T17:17:22Z Marlo Moesia 3113694 2843998 wikitext text/x-wiki = Successive Products Framework: Basic Principles = ''This learning resource is currently under development. You can find the main goals and an overview of the upcoming topics below.'' == Introduction == Welcome to this introductory course on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions, essential algebraic properties, and analytical tools required for practical implementation and application. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: The First Extension Based on the Fundamental Property''' * '''Subsequent Topics and Detailed Analysis''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\,\text{G}: \mathbb{R} \to \mathbb{R}\,</math> that takes as arguments another function <math>f(x),\,x \in \mathbb{R},\,</math> and a non-zero natural number <math>n,</math> such that <math> \text{G}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{G}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2)\,,\,\, </math> and so on. Thus, <math> \text{G}[f(x), x \leftarrow \{n\}] = f(1) \cdot f(2) \cdot f(3)\, \cdot ... \cdot f(n)\,;\,n\in\mathbb{N}^{*}. </math> From the expressions above, it becomes clear that <math>\operatorname{G}[f(x), x \leftarrow \{n\}] = \prod_{x=1}^{n} f(x); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> The function G is termed the 'Successive Product of <math>f(x)</math> at <math>x = n</math>'. As stated above, G is equivalent to a standard product starting at 1, provided that its computation is restricted to strictly positive natural numbers. ===Successive Products: Fundamental Property=== A standard product can be evaluated either directly, by performing all multiplications in one step, or recursively, by accumulating a partial result and multiplying it by each successive factor. This latter formulation allows the standard product at <math>n+1</math> to be written as <math>\prod_{x=1}^{n+1} f(x) = \left[\,\prod_{x=1}^{n} f(x)\right]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> Therefore, using successive product notation, one has <math>\operatorname{G}[f(x), x \leftarrow \{n+1\}] = \operatorname{G}[f(x), x \leftarrow \{n\}]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> which is termed the 'fundamental property of successive products'. ===Successive Products: The First Extension Based on the Fundamental Property=== A key aspect of the framework under consideration is that the fundamental property of successive products constitutes a recurrence relation; from this perspective, each value of G is computed from a preceding value. Consequently, there is no a priori reason to mandate that this recurrence must begin at <math>x = 1.</math> In fact, it could originate at 0, or even at earlier integer arguments, such as <math>-1, -2</math>, and so on, suggesting that no obstacles prevent extending the domain of the fundamental property of successive products from <math>\mathbb{N}^*</math> to <math>\mathbb{Z}\,</math>: <math>\operatorname{G}[f(x), x \leftarrow \{z+1\}] = \operatorname{G}[f(x), x \leftarrow \{z\}]\cdot f(z+1); \quad x \in \mathbb{R}, \, z \in \mathbb{Z}.</math> At this point, it becomes evident that the function G can no longer be represented by a standard product, because in any conventional product, the upper index must be greater than or equal to the lower index. Here, the usual products fail, and new rules governing operations between values of <math>f(x)</math> are established. These and related issues have been resolved. For example, <math>\operatorname{G}[f(x), x \leftarrow \{0\}] = 1\,,</math> is the only consistent result possible for G at <math>\,x=0</math>. ===Subsequent Topics and Detailed Analysis=== Marlo Moesia’s work proceeds from this point, addressing this and subsequent questions of interest, and extends successive products across the continuous domain of real numbers—extending definitions and properties from the realm of functions to mathematical operators that map real functions onto other real functions. This is an emerging field of research with compelling applications that has been successfully explored across several published works. A more detailed study can be pursued at https://moesia.org. <br> ---- lihq54pw7xews5xuxjgkw9fn4tvvpoa 2843999 2843998 2026-11-02T17:19:19Z Marlo Moesia 3113694 2843999 wikitext text/x-wiki = Successive Products Framework: Basic Principles = == Introduction == Welcome to this introductory course on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions, essential algebraic properties, and analytical tools required for practical implementation and application. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: The First Extension Based on the Fundamental Property''' * '''Subsequent Topics and Detailed Analysis''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\,\text{G}: \mathbb{R} \to \mathbb{R}\,</math> that takes as arguments another function <math>f(x),\,x \in \mathbb{R},\,</math> and a non-zero natural number <math>n,</math> such that <math> \text{G}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{G}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2)\,,\,\, </math> and so on. Thus, <math> \text{G}[f(x), x \leftarrow \{n\}] = f(1) \cdot f(2) \cdot f(3)\, \cdot ... \cdot f(n)\,;\,n\in\mathbb{N}^{*}. </math> From the expressions above, it becomes clear that <math>\operatorname{G}[f(x), x \leftarrow \{n\}] = \prod_{x=1}^{n} f(x); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> The function G is termed the 'Successive Product of <math>f(x)</math> at <math>x = n</math>'. As stated above, G is equivalent to a standard product starting at 1, provided that its computation is restricted to strictly positive natural numbers. ===Successive Products: Fundamental Property=== A standard product can be evaluated either directly, by performing all multiplications in one step, or recursively, by accumulating a partial result and multiplying it by each successive factor. This latter formulation allows the standard product at <math>n+1</math> to be written as <math>\prod_{x=1}^{n+1} f(x) = \left[\,\prod_{x=1}^{n} f(x)\right]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> Therefore, using successive product notation, one has <math>\operatorname{G}[f(x), x \leftarrow \{n+1\}] = \operatorname{G}[f(x), x \leftarrow \{n\}]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> which is termed the 'fundamental property of successive products'. ===Successive Products: The First Extension Based on the Fundamental Property=== A key aspect of the framework under consideration is that the fundamental property of successive products constitutes a recurrence relation; from this perspective, each value of G is computed from a preceding value. Consequently, there is no a priori reason to mandate that this recurrence must begin at <math>x = 1.</math> In fact, it could originate at 0, or even at earlier integer arguments, such as <math>-1, -2</math>, and so on, suggesting that no obstacles prevent extending the domain of the fundamental property of successive products from <math>\mathbb{N}^*</math> to <math>\mathbb{Z}\,</math>: <math>\operatorname{G}[f(x), x \leftarrow \{z+1\}] = \operatorname{G}[f(x), x \leftarrow \{z\}]\cdot f(z+1); \quad x \in \mathbb{R}, \, z \in \mathbb{Z}.</math> At this point, it becomes evident that the function G can no longer be represented by a standard product, because in any conventional product, the upper index must be greater than or equal to the lower index. Here, the usual products fail, and new rules governing operations between values of <math>f(x)</math> are established. These and related issues have been resolved. For example, <math>\operatorname{G}[f(x), x \leftarrow \{0\}] = 1\,,</math> is the only consistent result possible for G at <math>\,x=0</math>. ===Subsequent Topics and Detailed Analysis=== Marlo Moesia’s work proceeds from this point, addressing this and subsequent questions of interest, and extends successive products across the continuous domain of real numbers—extending definitions and properties from the realm of functions to mathematical operators that map real functions onto other real functions. This is an emerging field of research with compelling applications that has been successfully explored across several published works. A more detailed study can be pursued at https://moesia.org. <br> ---- s1jud94qa7grbpeek29kn0chb3va68o 2844000 2843999 2026-11-02T17:21:28Z Marlo Moesia 3113694 2844000 wikitext text/x-wiki = Successive Products Framework: Basic Principles = == Introduction == Welcome to this introductory topics on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions, essential algebraic properties, and analytical tools required for practical implementation and application. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: The First Extension Based on the Fundamental Property''' * '''Subsequent Topics and Detailed Analysis''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\,\text{G}: \mathbb{R} \to \mathbb{R}\,</math> that takes as arguments another function <math>f(x),\,x \in \mathbb{R},\,</math> and a non-zero natural number <math>n,</math> such that <math> \text{G}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{G}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2)\,,\,\, </math> and so on. Thus, <math> \text{G}[f(x), x \leftarrow \{n\}] = f(1) \cdot f(2) \cdot f(3)\, \cdot ... \cdot f(n)\,;\,n\in\mathbb{N}^{*}. </math> From the expressions above, it becomes clear that <math>\operatorname{G}[f(x), x \leftarrow \{n\}] = \prod_{x=1}^{n} f(x); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> The function G is termed the 'Successive Product of <math>f(x)</math> at <math>x = n</math>'. As stated above, G is equivalent to a standard product starting at 1, provided that its computation is restricted to strictly positive natural numbers. ===Successive Products: Fundamental Property=== A standard product can be evaluated either directly, by performing all multiplications in one step, or recursively, by accumulating a partial result and multiplying it by each successive factor. This latter formulation allows the standard product at <math>n+1</math> to be written as <math>\prod_{x=1}^{n+1} f(x) = \left[\,\prod_{x=1}^{n} f(x)\right]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> Therefore, using successive product notation, one has <math>\operatorname{G}[f(x), x \leftarrow \{n+1\}] = \operatorname{G}[f(x), x \leftarrow \{n\}]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> which is termed the 'fundamental property of successive products'. ===Successive Products: The First Extension Based on the Fundamental Property=== A key aspect of the framework under consideration is that the fundamental property of successive products constitutes a recurrence relation; from this perspective, each value of G is computed from a preceding value. Consequently, there is no a priori reason to mandate that this recurrence must begin at <math>x = 1.</math> In fact, it could originate at 0, or even at earlier integer arguments, such as <math>-1, -2</math>, and so on, suggesting that no obstacles prevent extending the domain of the fundamental property of successive products from <math>\mathbb{N}^*</math> to <math>\mathbb{Z}\,</math>: <math>\operatorname{G}[f(x), x \leftarrow \{z+1\}] = \operatorname{G}[f(x), x \leftarrow \{z\}]\cdot f(z+1); \quad x \in \mathbb{R}, \, z \in \mathbb{Z}.</math> At this point, it becomes evident that the function G can no longer be represented by a standard product, because in any conventional product, the upper index must be greater than or equal to the lower index. Here, the usual products fail, and new rules governing operations between values of <math>f(x)</math> are established. These and related issues have been resolved. For example, <math>\operatorname{G}[f(x), x \leftarrow \{0\}] = 1\,,</math> is the only consistent result possible for G at <math>\,x=0</math>. ===Subsequent Topics and Detailed Analysis=== Marlo Moesia’s work proceeds from this point, addressing this and subsequent questions of interest, and extends successive products across the continuous domain of real numbers—extending definitions and properties from the realm of functions to mathematical operators that map real functions onto other real functions. This is an emerging field of research with compelling applications that has been successfully explored across several published works. A more detailed study can be pursued at https://moesia.org. <br> ---- g0xbhsuns2pdk5jgc0bzmw6xga4vvdn 2844001 2844000 2026-11-02T17:56:41Z Marlo Moesia 3113694 /* Introduction */ 2844001 wikitext text/x-wiki = Successive Products Framework: Basic Principles = == Introduction == Welcome to this introductory topics on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions and essential algebraic properties. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: The First Extension Based on the Fundamental Property''' * '''Subsequent Topics and Detailed Analysis''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\,\text{G}: \mathbb{R} \to \mathbb{R}\,</math> that takes as arguments another function <math>f(x),\,x \in \mathbb{R},\,</math> and a non-zero natural number <math>n,</math> such that <math> \text{G}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{G}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2)\,,\,\, </math> and so on. Thus, <math> \text{G}[f(x), x \leftarrow \{n\}] = f(1) \cdot f(2) \cdot f(3)\, \cdot ... \cdot f(n)\,;\,n\in\mathbb{N}^{*}. </math> From the expressions above, it becomes clear that <math>\operatorname{G}[f(x), x \leftarrow \{n\}] = \prod_{x=1}^{n} f(x); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> The function G is termed the 'Successive Product of <math>f(x)</math> at <math>x = n</math>'. As stated above, G is equivalent to a standard product starting at 1, provided that its computation is restricted to strictly positive natural numbers. ===Successive Products: Fundamental Property=== A standard product can be evaluated either directly, by performing all multiplications in one step, or recursively, by accumulating a partial result and multiplying it by each successive factor. This latter formulation allows the standard product at <math>n+1</math> to be written as <math>\prod_{x=1}^{n+1} f(x) = \left[\,\prod_{x=1}^{n} f(x)\right]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> Therefore, using successive product notation, one has <math>\operatorname{G}[f(x), x \leftarrow \{n+1\}] = \operatorname{G}[f(x), x \leftarrow \{n\}]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> which is termed the 'fundamental property of successive products'. ===Successive Products: The First Extension Based on the Fundamental Property=== A key aspect of the framework under consideration is that the fundamental property of successive products constitutes a recurrence relation; from this perspective, each value of G is computed from a preceding value. Consequently, there is no a priori reason to mandate that this recurrence must begin at <math>x = 1.</math> In fact, it could originate at 0, or even at earlier integer arguments, such as <math>-1, -2</math>, and so on, suggesting that no obstacles prevent extending the domain of the fundamental property of successive products from <math>\mathbb{N}^*</math> to <math>\mathbb{Z}\,</math>: <math>\operatorname{G}[f(x), x \leftarrow \{z+1\}] = \operatorname{G}[f(x), x \leftarrow \{z\}]\cdot f(z+1); \quad x \in \mathbb{R}, \, z \in \mathbb{Z}.</math> At this point, it becomes evident that the function G can no longer be represented by a standard product, because in any conventional product, the upper index must be greater than or equal to the lower index. Here, the usual products fail, and new rules governing operations between values of <math>f(x)</math> are established. These and related issues have been resolved. For example, <math>\operatorname{G}[f(x), x \leftarrow \{0\}] = 1\,,</math> is the only consistent result possible for G at <math>\,x=0</math>. ===Subsequent Topics and Detailed Analysis=== Marlo Moesia’s work proceeds from this point, addressing this and subsequent questions of interest, and extends successive products across the continuous domain of real numbers—extending definitions and properties from the realm of functions to mathematical operators that map real functions onto other real functions. This is an emerging field of research with compelling applications that has been successfully explored across several published works. A more detailed study can be pursued at https://moesia.org. <br> ---- 8jw9ebke2vhcrz4hqsnnxns10858n21 2844055 2844001 2026-11-02T23:05:28Z Atcovi 276019 Atcovi moved page [[Moesia's Successive Operations Framework]] to [[User:Marlo Moesia/Moesia's Successive Operations Framework]] without leaving a redirect: moving to userspace 2844001 wikitext text/x-wiki = Successive Products Framework: Basic Principles = == Introduction == Welcome to this introductory topics on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions and essential algebraic properties. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: The First Extension Based on the Fundamental Property''' * '''Subsequent Topics and Detailed Analysis''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\,\text{G}: \mathbb{R} \to \mathbb{R}\,</math> that takes as arguments another function <math>f(x),\,x \in \mathbb{R},\,</math> and a non-zero natural number <math>n,</math> such that <math> \text{G}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{G}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2)\,,\,\, </math> and so on. Thus, <math> \text{G}[f(x), x \leftarrow \{n\}] = f(1) \cdot f(2) \cdot f(3)\, \cdot ... \cdot f(n)\,;\,n\in\mathbb{N}^{*}. </math> From the expressions above, it becomes clear that <math>\operatorname{G}[f(x), x \leftarrow \{n\}] = \prod_{x=1}^{n} f(x); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> The function G is termed the 'Successive Product of <math>f(x)</math> at <math>x = n</math>'. As stated above, G is equivalent to a standard product starting at 1, provided that its computation is restricted to strictly positive natural numbers. ===Successive Products: Fundamental Property=== A standard product can be evaluated either directly, by performing all multiplications in one step, or recursively, by accumulating a partial result and multiplying it by each successive factor. This latter formulation allows the standard product at <math>n+1</math> to be written as <math>\prod_{x=1}^{n+1} f(x) = \left[\,\prod_{x=1}^{n} f(x)\right]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> Therefore, using successive product notation, one has <math>\operatorname{G}[f(x), x \leftarrow \{n+1\}] = \operatorname{G}[f(x), x \leftarrow \{n\}]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> which is termed the 'fundamental property of successive products'. ===Successive Products: The First Extension Based on the Fundamental Property=== A key aspect of the framework under consideration is that the fundamental property of successive products constitutes a recurrence relation; from this perspective, each value of G is computed from a preceding value. Consequently, there is no a priori reason to mandate that this recurrence must begin at <math>x = 1.</math> In fact, it could originate at 0, or even at earlier integer arguments, such as <math>-1, -2</math>, and so on, suggesting that no obstacles prevent extending the domain of the fundamental property of successive products from <math>\mathbb{N}^*</math> to <math>\mathbb{Z}\,</math>: <math>\operatorname{G}[f(x), x \leftarrow \{z+1\}] = \operatorname{G}[f(x), x \leftarrow \{z\}]\cdot f(z+1); \quad x \in \mathbb{R}, \, z \in \mathbb{Z}.</math> At this point, it becomes evident that the function G can no longer be represented by a standard product, because in any conventional product, the upper index must be greater than or equal to the lower index. Here, the usual products fail, and new rules governing operations between values of <math>f(x)</math> are established. These and related issues have been resolved. For example, <math>\operatorname{G}[f(x), x \leftarrow \{0\}] = 1\,,</math> is the only consistent result possible for G at <math>\,x=0</math>. ===Subsequent Topics and Detailed Analysis=== Marlo Moesia’s work proceeds from this point, addressing this and subsequent questions of interest, and extends successive products across the continuous domain of real numbers—extending definitions and properties from the realm of functions to mathematical operators that map real functions onto other real functions. This is an emerging field of research with compelling applications that has been successfully explored across several published works. A more detailed study can be pursued at https://moesia.org. <br> ---- 8jw9ebke2vhcrz4hqsnnxns10858n21 2844057 2844055 2026-11-02T23:05:43Z Atcovi 276019 removing promotional link 2844057 wikitext text/x-wiki = Successive Products Framework: Basic Principles = == Introduction == Welcome to this introductory topics on successive products applied to functions <math>\,\text{F}: \mathbb{R} \to \mathbb{R}\,</math> and computed for integer arguments. This module provides a foundational framework, detailing primary definitions and essential algebraic properties. This material is designed as a foundational guide to help learners understand the general principles of this subtopic within the Moesia Successive Operations Framework. In this section, we will explore: * '''Successive Products: Definition Based on Usual Products''' * '''Successive Products: Fundamental Property''' * '''Successive Products: The First Extension Based on the Fundamental Property''' * '''Subsequent Topics and Detailed Analysis''' == Getting Started == While basic algebra is recommended, no prior background in this specific topic is required. === Successive Products: Definition Based on Usual Products === Consider a function <math>\,\text{G}: \mathbb{R} \to \mathbb{R}\,</math> that takes as arguments another function <math>f(x),\,x \in \mathbb{R},\,</math> and a non-zero natural number <math>n,</math> such that <math> \text{G}[f(x), x \leftarrow \{1\}] = f(1) </math> <math> \text{G}[f(x), x \leftarrow \{2\}] = f(1) \cdot f(2)\,,\,\, </math> and so on. Thus, <math> \text{G}[f(x), x \leftarrow \{n\}] = f(1) \cdot f(2) \cdot f(3)\, \cdot ... \cdot f(n)\,;\,n\in\mathbb{N}^{*}. </math> From the expressions above, it becomes clear that <math>\operatorname{G}[f(x), x \leftarrow \{n\}] = \prod_{x=1}^{n} f(x); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> The function G is termed the 'Successive Product of <math>f(x)</math> at <math>x = n</math>'. As stated above, G is equivalent to a standard product starting at 1, provided that its computation is restricted to strictly positive natural numbers. ===Successive Products: Fundamental Property=== A standard product can be evaluated either directly, by performing all multiplications in one step, or recursively, by accumulating a partial result and multiplying it by each successive factor. This latter formulation allows the standard product at <math>n+1</math> to be written as <math>\prod_{x=1}^{n+1} f(x) = \left[\,\prod_{x=1}^{n} f(x)\right]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> Therefore, using successive product notation, one has <math>\operatorname{G}[f(x), x \leftarrow \{n+1\}] = \operatorname{G}[f(x), x \leftarrow \{n\}]\cdot f(n+1); \quad x \in \mathbb{R}, \, n \in \mathbb{N}^*.</math> which is termed the 'fundamental property of successive products'. ===Successive Products: The First Extension Based on the Fundamental Property=== A key aspect of the framework under consideration is that the fundamental property of successive products constitutes a recurrence relation; from this perspective, each value of G is computed from a preceding value. Consequently, there is no a priori reason to mandate that this recurrence must begin at <math>x = 1.</math> In fact, it could originate at 0, or even at earlier integer arguments, such as <math>-1, -2</math>, and so on, suggesting that no obstacles prevent extending the domain of the fundamental property of successive products from <math>\mathbb{N}^*</math> to <math>\mathbb{Z}\,</math>: <math>\operatorname{G}[f(x), x \leftarrow \{z+1\}] = \operatorname{G}[f(x), x \leftarrow \{z\}]\cdot f(z+1); \quad x \in \mathbb{R}, \, z \in \mathbb{Z}.</math> At this point, it becomes evident that the function G can no longer be represented by a standard product, because in any conventional product, the upper index must be greater than or equal to the lower index. Here, the usual products fail, and new rules governing operations between values of <math>f(x)</math> are established. These and related issues have been resolved. For example, <math>\operatorname{G}[f(x), x \leftarrow \{0\}] = 1\,,</math> is the only consistent result possible for G at <math>\,x=0</math>. ===Subsequent Topics and Detailed Analysis=== Marlo Moesia’s work proceeds from this point, addressing this and subsequent questions of interest, and extends successive products across the continuous domain of real numbers—extending definitions and properties from the realm of functions to mathematical operators that map real functions onto other real functions. This is an emerging field of research with compelling applications that has been successfully explored across several published works.<br> ---- 9ktka82ax6pmb4f7qq4pi7yqgym8t7k 6 332778 2843992 2026-11-02T16:47:08Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20261102 - 20261027) |Source={{own|Young1lim}} |Date=2026-11-02 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2843992 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20261102 - 20261027) |Source={{own|Young1lim}} |Date=2026-11-02 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 1i0z3x3hcwrk0nselqbilf9xcnghtx9 6 332779 2843994 2026-11-02T17:07:56Z Young1lim 21186 {{Information |Description=Data.1A: Data Object (20261102 - 20261027) |Source={{own|Young1lim}} |Date=2026-11-02 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2843994 wikitext text/x-wiki == Summary == {{Information |Description=Data.1A: Data Object (20261102 - 20261027) |Source={{own|Young1lim}} |Date=2026-11-02 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 9g09dnoyd22c6bvq50884id8sryk3qc 0 332781 2844099 2026-11-03T04:06:12Z Jtneill 10242 Created page with "{{title|General feedback about<br>book chapters}} {{RoundBoxTop|theme=3}} * General feedback about the [[Motivation and emotion/Book/2025|2025 student-authored book chapters]] * Detailed feedback about each individual chapter is available on the respective talk pages {{RoundBoxBottom}} __TOC__ ==Overall== # The overall quality of chapters was reasonably good, but there was a wide range # The main issues were: ## Unacknowledged and/or poor use of genAI ## Insufficient r..." 2844099 wikitext text/x-wiki {{title|General feedback about<br>book chapters}} {{RoundBoxTop|theme=3}} * General feedback about the [[Motivation and emotion/Book/2025|2025 student-authored book chapters]] * Detailed feedback about each individual chapter is available on the respective talk pages {{RoundBoxBottom}} __TOC__ ==Overall== # The overall quality of chapters was reasonably good, but there was a wide range # The main issues were: ## Unacknowledged and/or poor use of genAI ## Insufficient review and synthesis of the best research about the topic ## Lack of social contributions <!-- # The best chapters will be [https://twitter.com/jtneill/status/1554447127639519232 tweeted in this thread] --> ==Overview== # The best overviews consisted of: ## A scenario/case study in a feature box with a relevant image ## A simple, evocative description of the psychological problem or phenomenon ## Clear and relevant focus questions in a feature box; the focus questions are closely aligned to the top-level headings. ==Theory== # Most chapters used a well selected range of relevant theories # Weaker chapters tended to use more general, historical theories; stronger chapters tended to identify more specific, recent theories # Theories were generally well explained, but theoretical depth and relevance was a key differentiator between chapters # All chapters could be improved by building more strongly on related chapters (i.e., by embedding links from the first mention of key words to related chapters) # More practical examples would have been useful to demonstrate how theories work in practice ==Research== # The best chapters provided comprehensive and critical synthesis of the best research # Most chapters review some relevant research, but could often the research could have been more relevant and/or indepth # Place more emphasis on major reviews such as meta-analyses # Sometimes there was insufficient citation of academic peer-reviewed sources to support claims ==Integration== # The best chapters had excellent integration between theory and research # Many chapters emphasised theory more than research — strive for balance ==Conclusion== # Useful summaries of key points were provided # The best chapters emphasised explicit take-home messages for each focus question ==Style== ===Written expression=== # Overall, the quality of written expression was reasonably good, but varied widely. # The most common problems were: ## grammar and spelling ### poor grammar — there's many ways to get help to develop professional-level written expression (e.g., Studiosity, genAI) ### use 3rd person, rather than 1st person, perspective[https://www.grammarly.com/blog/first-second-and-third-person/] ### use past tense when describing research studies, although implications of findings could be in the present tense ### [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should] should be used (they are part of APA style) ### use of [https://grammar.yourdictionary.com/punctuation/apostrophe-rules.html ownership apostrophes] ### Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling]. ## insufficient citation — scientific communication needs evidence to support claims ## clarity of written expression ### "People" is often a better term than "individuals" ### Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the content/key point to be communicated, with the citation included along the way or, more typically, in parentheses at the end of the sentence. # APA style: ## Citations with three or more authors should use the first author's surname followed by et al. and the year ## Reference formatting areas for improvement: ### capitalisation ### italicisation ### use of hyperlinked dois ===Learning features=== # Embedded interwiki links to Wikipedia articles help to connect reader to further information about key concepts # Embedded interwiki links to related Wikiversity [[Motivation and emotion/Book|book chapters]] were underutilised — network the chapter to related chapters # Figures were reasonably well used, with several students uploading their own images (thankyou!) # Tables were less commonly used, but usually very useful # Feature boxes were generally well used # Quizzes were generally well used # Scenarios/case studies were reasonably well used, but often more examples would have been helpful ==Social contributions== # There were many valuable improvements made to past and current chapters # The amount and quality of contributions varied widely — some students made no contributions whilst some excelled and were awarded bonus marks # Unless there were ''direct links'' to evidence, no marks were provided == See also == * [[Motivation and emotion/Assessment/Chapter/Feedback/2021|2021 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2022|2022 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2023|2023 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2024|2024 feedback]] * [[Motivation and emotion/Assessment/Chapter|Book chapter - Guidelines]]<!-- * [[Motivation and emotion/Assessment/Multimedia/Feedback|Multimedia presentation feedback]] --> [[Category:Motivation and emotion/Assessment/Chapter]] oz3e6d1d90h7poajlh6srrd1esvryac 3 332785 2844173 2026-11-03T10:52:28Z MathXplore 2888076 vandalism1 ([[m:User:ZbVl/VD|Vandoom]]) 2844173 wikitext text/x-wiki == 2026-11-03 == [[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Colloquium|Colloquium]]. Thank you.<!-- Glow-vandalism1 @ 1793703140948.5s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 10:52, 3 November 2026 (UTC) sv1u229tp44zr3m946cseiuq1861e0v